{"id":540,"date":"2017-11-01T00:36:45","date_gmt":"2017-10-31T19:06:45","guid":{"rendered":"http:\/\/ibelitetutor.com\/blog\/?p=540"},"modified":"2023-08-14T12:41:30","modified_gmt":"2023-08-14T07:11:30","slug":"algebra-tutors","status":"publish","type":"post","link":"http:\/\/ibelitetutor.com\/blog\/algebra-tutors\/","title":{"rendered":"Permutations and Combinations-algebra tutors"},"content":{"rendered":"

Permutations and Combinations(part-2)<\/span><\/h2>\n

In the previous post, IB Maths Tutors<\/a> <\/strong>discussed the fundamental principle of counting and various methods of permutations<\/strong>.\u00a0In this post, I shall discuss combinations in details.<\/p>\n

Meaning of Combination<\/strong><\/span><\/h3>\n

If we are given a set of objects and we want to select\u00a0a few objects out of this set, then we can do it by many different ways. These ways are known as combinations.
\nExample-<\/strong> <\/span>If we are given three balls marked as B, W and R and we want to select two balls then we can select like this- BW, BR, WR.<\/p>\n

These are known as the combination of this selection. This is an important concept in Online Maths Tutoring<\/strong><\/p>\n

Combination of n different objects taken r at a time when repetition is not allowed<\/strong><\/span><\/h3>\n

If repetition is not allowed the number of ways of selecting r objects out of a group of n objects is called \u00a0\"{}^n{c_r}\"<\/p>\n

\"{}^n{c_r}\"= \u00a0\u00a0\"\\frac{{n!}}{{r!\\left(<\/p>\n

In latest notation system\u00a0\u00a0\"{}^n{c_r}\"\u00a0is also known as C(n;r) or \u00a0\u00a0\"\\left(<\/p>\n

Properties of \u00a0<\/span>\"\\left(<\/strong><\/h3>\n

It’s a very useful and interesting Mathematical tool. It has following properties.<\/p>\n

(i)\u00a0\"{}^n{c_r}\"=\u00a0\"{}^n{c_{n<\/p>\n

(ii) \u00a0\"{}^n{c_n}<\/p>\n

(iii) \u00a0\"{}^n{c_r}\u00a0 \u00a0 known as Pascal’s law<\/p>\n

(iv) r.\u00a0\"{}^n{c_r}<\/p>\n

(v) \u00a0\"\\frac{{{}^n{c_r}}}{{{}^n{c_{r<\/p>\n

(vi) If n is even then we should put r=n\/2 for maximum value of\u00a0\u00a0\"{}^n{c_r}\"\u00a0and if n is odd then\u00a0\u00a0\"{}^n{c_r}\"\u00a0is greatest when r=\u00a0\"\\frac{{{n^2}<\/p>\n

(vii) In the expansions of\u00a0\"{(1\u00a0if we put x=1 then<\/p>\n

\"{}^n{c_0} <\/p>\n

\"{}^n{c_0} <\/p>\n

\"{}^n{c_1} <\/p>\n

Some difficult Combinations<\/strong><\/span><\/h4>\n

(a) If we want to select r objects out of a set of n objects such that p particular objects are always selected then the total number of ways = \u00a0\"{}^{n<\/p>\n

(b) If we want to select r objects out of a set of n objects such that p particular objects are never selected then the total number of ways = \u00a0\"{}^{n<\/p>\n

(c)\u00a0If we want to select one or more object(at least one object) out of a set of n objects then the total number of ways=\u00a0\"{2^n}<\/p>\n

(d)\u00a0If we want to select zero or more object(at least zero object) out of a set of n objects then the total number of ways= \u00a0\"{2^n}\"<\/p>\n

(e) If we want to select two or more object(at least two objects) out of a set of n objects then the total number of ways= \u00a0\"{2^n}<\/p>\n

A few more Combinations….<\/span><\/h4>\n

(f) If we want to select one or more object(at least one object) out of a set of n identical objects then the total number of ways=\u00a0\"{}^n{c_1}\"\u00a0= n<\/p>\n

(g)\u00a0If we have a set of objects in which p objects are of one type, q objects are of other type\u00a0and r objects are of some other type and we want to select some or all out of this set then the total number of combinations=(p+1)(q+1)(r+1)-1<\/p>\n

(h)\u00a0If we have a set of objects in which p objects are of one type, q objects are of other type, r objects are of some other type, n objects are different and we want to select at least one out of this set then total number of combinations=(p+1)(q+1)(r+1)\u00a0\"{2^n}\"<\/p>\n

(i) If we have a set of objects in which p objects are of one type, q objects are of other type, r objects are of other type, n objects are different and we want to select such that at least one object of each kind is included total number of combinations= pqr(\u00a0\"{2^n})<\/p>\n

(j) Number of ways in which (m+n+p) can be divided into three unequal groups=\"\\frac{{(m<\/p>\n

(h) Number of ways in which (m+n+p) can be divided and distributed into three unequal groups=\u00a0\"\\frac{{(m<\/p>\n

Problems where we use both permutations and combinations<\/strong><\/span><\/h3>\n

In exams, usually, we are asked questions in which we have to use both permutations and combinations.<\/p>\n

(a) The number of ways of selecting and arranging r objects out of a group of n different objects such that p particular objects are always included=\u00a0\"{}^{n.\u00a0r!<\/p>\n

(b)\u00a0The number of ways of selecting and arranging r objects out of a group of n different objects such that p particular objects are always excluded = \u00a0\"{}^{n.r!<\/p>\n

(c) The number of ways of selecting and arranging r objects out of a group of n different objects such that p particular objects are always seperated= \u00a0\"{}^{n<\/p>\n

De-arrangement theorem<\/strong><\/span><\/h4>\n

This theorem is used to permutate n distinct objects such that no object gets its original position.<\/p>\n

according to this theorem, total no. of ways= \u00a0\"n!\\left[<\/p>\n

Formation of Geometrical Figures Using Combinations-<\/p>\n

(a) If we are given n points on a plane then, the number of lines formed by joining these points = \u00a0\"{}^n{c_2}\"<\/p>\n

(b) If we are given n points on a plane out of which m are in a straight line, the number of lines formed by joining these points=\u00a0\"{}^n{c_2}<\/p>\n

(c) If we are given n points on a plane then, the number of triangles formed by joining these points = \u00a0\"{}^n{c_3}\"<\/p>\n

(d) If we are given n points on a plane out of which m are in a straight line, the number of triangles formed by joining these points= \u00a0\"{}^n{c_3}<\/p>\n

(e) If we are given n points on a plane, the number of diagonals formed by joining these points=\u00a0\"{}^n{c_2}\u00a0= \u00a0\"\\frac{{n(n<\/p>\n

Here I am posting a pdf. Use concepts that are given in this post and the previous post of permutation and combination<\/strong> to solve questions given in the pdf<\/p>\n

\"ib<\/p>\n

For free Online class. Whatsapp at +919911262206 or fill the form<\/strong><\/h5>\n\n
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    <\/p>\n","protected":false},"excerpt":{"rendered":"

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