By K. M. Koh, Tay Eng Guan, Eng Guan Tay

ISBN-10: 9812380639

ISBN-13: 9789812380630

ISBN-10: 9812380647

ISBN-13: 9789812380647

ISBN-10: 9812777121

ISBN-13: 9789812777126

Presents an invaluable, beautiful advent to uncomplicated counting strategies for top secondary and junior students, in addition to lecturers. is helping scholars get an early begin to studying problem-solving heuristics and pondering abilities.

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**Extra info for Counting MSch**

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N! 6 ) = 6S! 9 4 ! = H92052400. (5-2)! 3! V / Note that when r = 0 or n, we have ; - i - ; , . i . Again, by convention, we define = 1. 6) Thus, (™) = («>) = 45 and Q = ^f) = 1192052400. Subsets and Arrangements 23 We define P™ as the number of r-permutations and (") as the number of r-element subsets of N n . Actually, in these definitions, N n can be replaced by any n-element set since it is the number of the elements but not the nature of the elements in the set that matters. That is, given any n-element set S, P™ (respectively, (")) is defined as the number of r-permutations (respectively, r-element subsets) of S.

1). This correspondence clearly establishes a bijection between the set of nonnegative integer solutions to (1) and the set of ways of distributing 7 identical balls in 3 distinct boxes. 2), we can actually establish the following general results. Consider the linear equation x\ + a>2 H \-xn = r (2) where r is a nonnegative integer. 1) (i) The number of nonnegative integer solutions to (2) is given by (r+^1). , xn) to (2), with each Xj > 1, is given by (£Z„)> where r > n and i = 1,2,... ,n. 2 Recall that the number of 3-element subsets {a, b, c} of the set N 1 0 = { 1 , 2 , 3 , .

3-3! + --- + n - n ! This page is intentionally left blank Chapter 4 Applications Having introduced the concepts of r-permutations and r-combinations of an n-element set, and having derived the formulae for P? and (™), we shall now give some examples to illustrate how these can be applied. 1 There are 6 boys and 5 men waiting for their turn in a barber shop. Two particular boys are A and B, and one particular man is Z. There is a row of 11 seats for the customers. Find the number of ways of arranging them in each of the following cases: (i) there are no restrictions; (ii) A and B are adjacent; (iii) Z is at the centre, A at his left and B at his right (need not be adjacent); (iv) boys and men alternate.

### Counting MSch by K. M. Koh, Tay Eng Guan, Eng Guan Tay

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