{"id":532,"date":"2017-10-28T00:53:34","date_gmt":"2017-10-27T19:23:34","guid":{"rendered":"http:\/\/ibelitetutor.com\/blog\/?p=532"},"modified":"2025-05-27T00:46:49","modified_gmt":"2025-05-26T19:16:49","slug":"permutations-and-combinations","status":"publish","type":"post","link":"http:\/\/ibelitetutor.com\/blog\/permutations-and-combinations\/","title":{"rendered":"Permutation and Combination"},"content":{"rendered":"<h2><span style=\"color: #0000ff;\"><strong>Permutations and Combinations<\/strong><\/span><\/h2>\n<p>&#8216;<strong>Permutations and Combinations<\/strong>&#8216;\u00a0is the next post of my series <strong>Online Maths Tutoring<\/strong>. It is very useful and interesting as a topic. It&#8217;s also very useful in solving problems of Probability. Our<strong><a href=\"https:\/\/ibelitetutor.com\/ib-maths-tutors\/\"> IB Maths Tutors<\/a><\/strong> say that to understand Permutations and Combinations, we first need to understand Factorial.<\/p>\n<p><span style=\"color: #ff6600;\"><strong>Definition\u00a0of Factorial-<\/strong><\/span><\/p>\n<p>If we multiply n consecutive natural numbers together, then the product is called factorial of n. Its shown by n! or by<\/p>\n<p>for example : \u00a0 \u00a0 \u00a0\u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=n%21%20%3D%20n%28n%20-%201%29%28n%20-%202%29%28n%20-%203%29..........3.2.1\" alt=\"n! = n(n - 1)(n - 2)(n - 3)..........3.2.1\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacg cacqGH9aqpcaWGUbGaaiikaiaad6gacqGHsislcaaIXaGaaiykaiaa cIcacaWGUbGaeyOeI0IaaGOmaiaacMcacaGGOaGaamOBaiabgkHiTi aaiodacaGGPaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGa aiOlaiaac6cacaGGUaGaaiOlaiaaiodacaGGUaGaaGOmaiaac6caca aIXaaaaa!4FF1! \" \/><\/p>\n<h4><span style=\"color: #ff6600;\"><strong>Some Properties of Factorials<\/strong><\/span><\/h4>\n<p>(i) Factorials can only be calculated for positive integers at this level. We use gamma functions to define non-integer factorial\u00a0that&#8217;s not required at this level<\/p>\n<p>(ii) Factorial of a number can be written as a product of that number with the factorial of its predecessor \u00a0 \u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=n%21%20%3D%20n%5B%28n%20-%201%29%28n%20-%202%29%28n%20-%203%29..........3.2.1%5D\" alt=\"n! = n[(n - 1)(n - 2)(n - 3)..........3.2.1]\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacg cacqGH9aqpcaWGUbGaai4waiaacIcacaWGUbGaeyOeI0IaaGymaiaa cMcacaGGOaGaamOBaiabgkHiTiaaikdacaGGPaGaaiikaiaad6gacq GHsislcaaIZaGaaiykaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGa aiOlaiaac6cacaGGUaGaaiOlaiaac6cacaaIZaGaaiOlaiaaikdaca GGUaGaaGymaiaac2faaaa!51B1! \" \/><\/p>\n<p><img decoding=\"async\" class=\"ee_img tr_noresize\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%20%3D%20n%28n%20-%201%29%21\" alt=\" = n(n - 1)!\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam OBaiaacIcacaWGUbGaeyOeI0IaaGymaiaacMcacaGGHaaaaa!3C87! \" \/><\/p>\n<p>(iii) \u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=0%21%20%3D%201\" alt=\"0! = 1\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaacg cacqGH9aqpcaaIXaaaaa!3915! \" \/>\u00a0 you can watch this video for the explanation.<\/p>\n<p><iframe title=\"Value of factorial zero\" width=\"900\" height=\"506\" src=\"https:\/\/www.youtube.com\/embed\/SKp0QIDtBFE?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<p><!--more-->(iv) \u00a0If we want to simplify a &#8220;permutations and combinations&#8221; expression that has factorials in the numerator as well as in the denominator, we make all the factorials equal to the smallest factorial<\/p>\n<h3><span style=\"color: #ff6600;\"><strong>Exponent o<span style=\"color: #ff6600;\">f Prime Number p in n!<\/span><\/strong><\/span><\/h3>\n<p>Let&#8217;s assume that p is a prime\u00a0number and n is a positive integer, then exponent of p in n! is denoted by \u00a0E<sub>p <\/sub>(n!)<\/p>\n<p><!--StartFragment --> <img decoding=\"async\" class=\"ee_img tr_noresize\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%7BE_p%7D%28n%21%29%20%3D%20%5Cleft%5B%20%7B%5Cfrac%7Bn%7D%7Bp%7D%7D%20%5Cright%5D%20%2B%20%5Cleft%5B%20%7B%5Cfrac%7Bn%7D%7B%7B%7Bp%5E2%7D%7D%7D%7D%20%5Cright%5D%20%2B%20........%5Cleft%5B%20%7B%5Cfrac%7Bn%7D%7B%7B%7Bp%5Et%7D%7D%7D%7D%20%5Cright%5D\" alt=\"{E_p}(n!) = \\left[ {\\frac{n}{p}} \\right] + \\left[ {\\frac{n}{{{p^2}}}} \\right] + ........\\left[ {\\frac{n}{{{p^t}}}} \\right]\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGWbaabeaakiaacIcacaWGUbGaaiyiaiaacMcacqGH9aqp daWadaqaamaalaaabaGaamOBaaqaaiaadchaaaaacaGLBbGaayzxaa Gaey4kaSYaamWaaeaadaWcaaqaaiaad6gaaeaacaWGWbWaaWbaaSqa beaacaaIYaaaaaaaaOGaay5waiaaw2faaiabgUcaRiaac6cacaGGUa GaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaWaamWaaeaadaWc aaqaaiaad6gaaeaacaWGWbWaaWbaaSqabeaacaWG0baaaaaaaOGaay 5waiaaw2faaaaa!5116! \" \/><\/p>\n<p>We can&#8217;t use this result to find the exponent of composite numbers.<\/p>\n<h3><span style=\"color: #ff6600;\"><strong>Fundamental Principle of Counting<\/strong><\/span><\/h3>\n<p>Almost all <strong>IB Online Tutors<\/strong>, teach the first exercise of Permutations and Combinations that is based on\u00a0the Fundamental Principle of Counting. We can learn it in two steps.<\/p>\n<h3><span style=\"color: #ff6600;\"><strong>Principle of Addition<\/strong><\/span><\/h3>\n<p><strong>\u00a0<\/strong>If there are x different ways to do a work and y different ways to two another work and both the works are independent of each other then there are<strong> (x+y)<\/strong> ways to do either first <span style=\"color: #ff0000;\">OR<\/span> second work<\/p>\n<h4><span style=\"color: #ff6600;\"><strong>Example-<\/strong>\u00a0<\/span><\/h4>\n<p>If we can choose a man in a team by 6 different ways and a woman by 4 different ways then we can choose either a man or a woman by 6+4=10 different ways.<\/p>\n<h4><span style=\"color: #ff6600;\"><strong>The principle of Multiplication<\/strong><\/span><\/h4>\n<p>If there are x different ways to do a work and y different ways to do another work and both the works are independent of each other then there are (x.y) ways to do both first <span style=\"color: #ff0000;\">AN<span style=\"color: #ff0000;\">D<\/span><\/span> second works.<\/p>\n<h4><span style=\"color: #ff6600;\"><strong>Example<\/strong><\/span><\/h4>\n<p>If we can choose a man in a team by 6 different ways and a woman by 4 different ways then we can choose a man and a woman by 6*4=24 different ways.<\/p>\n<h4><span style=\"color: #ff6600;\"><strong>Definition of Permutation<\/strong><\/span><\/h4>\n<p>The process of making different arrangements of objects, letters and words etc by changing their position is known as permutation<\/p>\n<h4><span style=\"color: #ff6600;\"><strong>Example<\/strong><\/span><\/h4>\n<p>A, B, and C are four books then we can arrange them BY 6 DIFFERENT WAYS \u00a0ABC, ACB BCA, BAC\u00a0CAB CBA. so we can say that there are 6 different permutations of this arrangement.<\/p>\n<h4><span style=\"color: #ff6600;\"><strong>Number of Permutations of n different objects taken all at a time<\/strong><\/span><\/h4>\n<p>If we want to arrange n objects at n different places then the total number of ways of doing this or the total number of permutations =\u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%7B%7D%5En%7Bp_n%7D\" alt=\"{}^n{p_n}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWraaSqabe aacaWGUbaaaOGaamiCamaaBaaaleaacaWGUbaabeaaaaa!3934! \" \/><\/p>\n<p>=n! here P represents permutations<\/p>\n<h4><span style=\"color: #ff6600;\"><strong>Number of Permutations of n different objects taken r at a time<\/strong><\/span><\/h4>\n<p><strong>\u00a0<\/strong>If we want to arrange n objects at r different places then the total number of ways of doing this or the total number of permutations = \u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%7B%7D%5En%7Bp_r%7D\" alt=\"{}^n{p_r}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWraaSqabe aacaWGUbaaaOGaamiCamaaBaaaleaacaWGYbaabeaaaaa!3938! \" \/><\/p>\n<p>= \u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%5Cfrac%7B%7Bn%21%7D%7D%7B%7B%5Cleft%28%20%7Bn%20-%20r%7D%20%5Cright%29%21%7D%7D\" alt=\"\\frac{{n!}}{{\\left( {n - r} \\right)!}}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGUbGaaiyiaaqaamaabmaabaGaamOBaiabgkHiTiaadkhaaiaawIca caGLPaaacaGGHaaaaaaa!3CA2! \" \/>\u00a0 \u00a0\u00a0here \u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%7B%7D%5En%7Bp_r%7D\" alt=\"{}^n{p_r}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWraaSqabe aacaWGUbaaaOGaamiCamaaBaaaleaacaWGYbaabeaaaaa!3938! \" \/>\u00a0represent permutations of n objects taken r at a time.<\/p>\n<h4><span style=\"color: #ff6600;\"><strong>Number of Permutations of n \u00a0objects when all objects are not\u00a0different<\/strong><\/span><\/h4>\n<p><strong>\u00a0<\/strong>If we have n objects in total out of which p are of one type, q are of another type, r are of any other type, remaining objects are all different from each other, the total number of ways of arranging them= \u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%5Cfrac%7B%7Bn%21%7D%7D%7B%7Bp%21q%21r%21%7D%7D\" alt=\"\\frac{{n!}}{{p!q!r!}}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGUbGaaiyiaaqaaiaadchacaGGHaGaamyCaiaacgcacaWGYbGaaiyi aaaaaaa!3C6E! \" \/><\/p>\n<p><span style=\"color: #ff6600;\"><strong>Number of Permutations of n different objects taken all at a time when repetition of objects is allowed<\/strong><\/span><\/p>\n<p>If we want to arrange n objects at n different places and we are free to repeat objects as many times as we wish, then the total number of ways of doing this or the total number of permutations =\u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%7Bn%5En%7D\" alt=\"{n^n}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaCa aaleqabaGaamOBaaaaaaa!3808! \" \/><\/p>\n<p><span style=\"color: #0000ff;\"><strong>Number of Permutations of n different objects taken r at a time when repetition of objects is allowed<\/strong><\/span><\/p>\n<p><strong>\u00a0<\/strong>If we want to arrange n objects at r different places(taking r at a time) and we are free to repeat objects as many times as we wish, then the total number of ways of doing this or the total number of permutations=\u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%7Bn%5Er%7D\" alt=\"{n^r}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaCa aaleqabaGaamOCaaaaaaa!380C! \" \/><\/p>\n<p><strong>Circular Permutations-<\/strong> When we talk about arrangements of objects, it usually means linear arrangements. But if we wish, we can also arrange objects in a loop. Like we can ask our guests to sit around a round dining table. These types of arrangements are called circular permutations.<\/p>\n<p>If we want to arrange n objects in a circle, then the total number of ways\/circular permutations=(n-1)! this case works when there is<span style=\"color: #ff0000;\"> some difference<\/span> between clock-wise and anti-clockwise orders<\/p>\n<p>IF there is <span style=\"color: #ff0000;\">no distinction<\/span> between clock-wise and anti-clockwise orders, the total number of permutations=(n-1)!\/2<\/p>\n<p>Take help from <strong><a href=\"https:\/\/ibelitetutor.com\/ib-maths-tutors\/\">Maths Tutors <\/a><\/strong>for free in case of any difficulty<\/p>\n<h4><span style=\"color: #0000ff;\"><strong>Restricted Permutations<\/strong><\/span><\/h4>\n<p>There may be following cases of restricted permutation<\/p>\n<p>(a)\u00a0\u00a0\u00a0Number of arrangements of \u2018n\u2019 objects, taken \u2018r\u2019 at a time, when a particular object is to be always included \u00a0 \u00a0 =\u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=r%7B.%5E%7Bn%20-%201%7D%7D%7BP_%7Br%20-%201%7D%7D\" alt=\"r{.^{n - 1}}{P_{r - 1}}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGYbGaaiiOaiaac6capaWaaWbaaSqabeaapeGaamOBaiabgkHi TiaaigdaaaGccaGGGcGaamiua8aadaWgaaWcbaWdbiaadkhacqGHsi slcaaIXaaapaqabaaaaa!40C5! \" \/><\/p>\n<p>(b)\u00a0Number of arrangements of \u2018n\u2019 objects, taken \u2018r\u2019 at a time, when a particular object is fixed: = \u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%5E%7Bn%20-%201%7D%7BP_%7Br%20-%201%7D%7D\" alt=\"^{n - 1}{P_{r - 1}}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWbaaSqabe aaqaaaaaaaaaWdbiaad6gacqGHsislcaaIXaaaaOGaaiiOaiaadcfa paWaaSbaaSqaa8qacaWGYbGaeyOeI0IaaGymaaWdaeqaaaaa!3DD9! \" \/><\/p>\n<p>(c)\u00a0The number of arrangements of \u2018n\u2019 objects, taken \u2018r\u2019 at a time, when a particular object is never taken: =\u00a0<sup>n-1<\/sup>\u00a0P<sub>r.<\/sub><\/p>\n<p>(d)\u00a0The number of arrangements of \u2018n\u2019 objects, taken \u2018r\u2019 at a time, when \u2018m\u2019 specific objects always come with each-other = \u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=m%21%7B%5Crm%7B%20%7D%7D%7B%5Crm%7B.%7D%7D%5Cleft%28%20%7B%7B%5Crm%7B%20%7D%7Dn%20-%20m%20%2B%201%7D%20%5Cright%29%7B%5Crm%7B%20%7D%7D%21\" alt=\"m!{\\rm{ }}{\\rm{.}}\\left( {{\\rm{ }}n - m + 1} \\right){\\rm{ }}!\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGTbGaaiyiaiaacckacaqGGaGaaeOla8aadaqadaqaa8qacaGG GcGaaeiiaiaad6gacqGHsislcaWGTbGaey4kaSIaaGymaaWdaiaawI cacaGLPaaapeGaaeiiaiaacgcaaaa!4369! \" \/><\/p>\n<p>(e)\u00a0The number of arrangements of \u2018n\u2019 things, taken all at a time, when \u2018m\u2019 specific objects always come with each other=\u00a0<img decoding=\"async\" class=\"ee_img tr_noresize\" style=\"font-size: 0.95em;\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%5Cbegin%7Barray%7D%7Blllllllllllllll%7D%0A%7Bn%7B%5Crm%7B%20%7D%7D%21%7B%5Crm%7B%20%7D%7D%20-%20%7B%5Crm%7B%20%7D%7D%5Cleft%5B%20%7B%7B%5Crm%7B%20%7D%7Dm%21%5Cleft%28%20%7Bn%20-%20m%20%2B%201%7D%20%5Cright%29%21%7B%5Crm%7B%20%7D%7D%7D%20%5Cright%5D%7D%5C%5C%0A%0A%5Cend%7Barray%7D\" alt=\"\\begin{array}{lllllllllllllll} {n{\\rm{ }}!{\\rm{ }} - {\\rm{ }}\\left[ {{\\rm{ }}m!\\left( {n - m + 1} \\right)!{\\rm{ }}} \\right]}\\\\ \\end{array}\" longdesc=\"MathType!MTEF!2!1!+- feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaa qaaabaaaaaaaaapeGaamOBaiaabccacaGGHaGaaeiiaiabgkHiTiaa bccapaWaamWaaeaapeGaaeiiaiaad2gacaGGHaWdamaabmaabaWdbi aad6gacqGHsislcaWGTbGaey4kaSIaaGymaaWdaiaawIcacaGLPaaa peGaaiyiaiaabccaa8aacaGLBbGaayzxaaaabaWdbiaacckaaaaaaa!479B! \" \/><\/p>\n<p>In my next post, I will discuss in detail about combinations and will share a large worksheet based on \u00a0P &amp; C. In the meantime you can download and solve these questions.<\/p>\n<p><span style=\"color: #0000ff;\"><strong>Also, Check the below-given post on Permutation and Combination<\/strong><\/span><\/p>\n<blockquote class=\"wp-embedded-content\" data-secret=\"iMEFWeKh2V\"><p><a href=\"https:\/\/ibelitetutor.com\/blog\/algebra-tutors\/\">Permutations and Combinations-algebra tutors<\/a><\/p><\/blockquote>\n<p><iframe class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; clip: rect(1px, 1px, 1px, 1px);\" title=\"&#8220;Permutations and Combinations-algebra tutors&#8221; &#8212; IB Elite Tutor\" src=\"https:\/\/ibelitetutor.com\/blog\/algebra-tutors\/embed\/#?secret=1s9jePkokr#?secret=iMEFWeKh2V\" data-secret=\"iMEFWeKh2V\" width=\"600\" height=\"338\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe><\/p>\n<p><strong>Click to <\/strong>getting<strong> FREE<\/strong> <a href=\"https:\/\/ibelitetutor.com\/ib-maths-tutors\/\"><strong>Online Maths Tutoring\u00a0<\/strong><\/a><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-931\" src=\"http:\/\/ibelitetutor.com\/blog\/wp-content\/uploads\/2018\/04\/ib-free-demo-class.png\" alt=\"ib free demo class\" width=\"300\" height=\"169\" \/><\/p>\n<p><strong>Whatsapp at +919911262206 or fill the form to get 1 hr free Class<\/strong><\/p>\n\n<div class=\"wpcf7 no-js\" id=\"wpcf7-f168-o1\" lang=\"en-US\" dir=\"ltr\" data-wpcf7-id=\"168\">\n<div class=\"screen-reader-response\"><p role=\"status\" aria-live=\"polite\" aria-atomic=\"true\"><\/p> <ul><\/ul><\/div>\n<form action=\"\/blog\/wp-json\/wp\/v2\/posts\/532#wpcf7-f168-o1\" method=\"post\" class=\"wpcf7-form init\" aria-label=\"Contact form\" novalidate=\"novalidate\" data-status=\"init\">\n<fieldset class=\"hidden-fields-container\"><input type=\"hidden\" name=\"_wpcf7\" value=\"168\" \/><input type=\"hidden\" name=\"_wpcf7_version\" value=\"6.1.5\" \/><input type=\"hidden\" name=\"_wpcf7_locale\" value=\"en_US\" \/><input type=\"hidden\" name=\"_wpcf7_unit_tag\" value=\"wpcf7-f168-o1\" \/><input type=\"hidden\" name=\"_wpcf7_container_post\" value=\"0\" \/><input type=\"hidden\" name=\"_wpcf7_posted_data_hash\" value=\"\" \/>\n<\/fieldset>\n<p><label> Your Email (required)<br \/>\n<span class=\"wpcf7-form-control-wrap\" data-name=\"your-email\"><input size=\"40\" maxlength=\"400\" class=\"wpcf7-form-control wpcf7-email wpcf7-validates-as-required wpcf7-text wpcf7-validates-as-email\" aria-required=\"true\" aria-invalid=\"false\" value=\"\" type=\"email\" name=\"your-email\" \/><\/span> <\/label>\n<\/p>\n<p><label> Your Message with Whatsapp number<br \/>\n<span class=\"wpcf7-form-control-wrap\" data-name=\"your-subject\"><input size=\"40\" maxlength=\"400\" class=\"wpcf7-form-control wpcf7-text\" aria-invalid=\"false\" value=\"\" type=\"text\" name=\"your-subject\" \/><\/span> <\/label><br \/>\n<span class=\"wpcf7-form-control-wrap\" data-name=\"quiz-math\"><label><span class=\"wpcf7-quiz-label\">6+5=?<\/span> <input size=\"40\" class=\"wpcf7-form-control wpcf7-quiz quiz\" autocomplete=\"off\" aria-required=\"true\" aria-invalid=\"false\" type=\"text\" name=\"quiz-math\" \/><\/label><input type=\"hidden\" name=\"_wpcf7_quiz_answer_quiz-math\" value=\"4b4fb63a4306aaeec18ee5bbc5dc11ad\" \/><\/span>\n<\/p>\n<p><input class=\"wpcf7-form-control wpcf7-submit has-spinner\" type=\"submit\" value=\"Send\" \/>\n<\/p><div class=\"wpcf7-response-output\" aria-hidden=\"true\"><\/div>\n<\/form>\n<\/div>\n\n<p><!--EndFragment --><\/p>\n<p><!--EndFragment --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Permutations and Combinations &#8216;Permutations and Combinations&#8216;\u00a0is the next post of my series Online Maths Tutoring. It is very useful and interesting as a topic. 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