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

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

Offers an invaluable, appealing creation to simple counting innovations for higher secondary and junior students, in addition to academics. is helping scholars get an early begin to studying problem-solving heuristics and pondering abilities.

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3,2,1; that is, n! = n ( n - l ) ( n - 2 ) . . 2) Thus 4! = 4 • 3 • 2 • 1 = 24. " is read "n factorial". By convention, we define 0! = 1. Using the "factorial" notation, we now have P r n = n(n - 1 ) . . (n - r + 1) n{n — 1 ) . . (n — r + l)(n — r)(n — r — 1 ) . . 2 • 1 _ n\ (n — r)(n — r — 1 ) . . 3) When n = 4 and r = 3, we obtain 4! r 3 — (4 - 3)! P4 - 4! 1! 4• 3• 2• 1 = 4 • 3 • 2 = 24, 1 which agrees with what we found before. 3) is valid when 0 < r < n. Consider two extreme cases: when r = 0 and r = n respectively.

It does not matter how different the members in A and those in B are in nature. As long as there exists a bijection between them, we get |vl| = |B|. 1 (a) Find the number of positive divisors of n if (i) n = 31752; (ii) n = 55125. 2 (b) In general, given an integer n > 2, how do you find the number of positive divisors of re? 5 the the the the routes must routes must routes must segment AB pass through A; pass through AB; pass through A and C; is deleted. For each positive integer n, show that ^ ( N ^ l = 2" by establishing a bijection between V(Nn) and the set of n-digit binary sequences.

Since 4, 1, 2 are nonnegative integers, we say that (xi,X2,xz) = (4,1,2) is a nonnegative integer solution to the linear equation (1). Note that (x\,X2,xs) = (1,2,4) is also a nonnegative integer solution to (1), and so are (4, 2, 1) and (1, 4, 2). Other nonnegative integer solutions to (1) include (0,0,7), (0,7,0), (1,6,0), (5,1,1),... 1 Find the number of nonnegative integer solutions to (1). Solution Let us create 3 distinct "boxes" to represent xi,x2 and £3, respectively. 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.

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