By K. M. Koh, Tay Eng Guan, Eng Guan Tay
Presents an invaluable, appealing creation to easy counting concepts for higher secondary and junior students, in addition to academics. is helping scholars get an early begin to studying problem-solving heuristics and considering abilities.
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Extra resources for Counting MSch
In this chapter, we shall introduce another basic principle for counting which we call the Bijection Principle, and discuss some of its applications. 1 Suppose that there are 200 parking lots in a multi-storey carpark. 1). Then we know that the number of vehicles in the carpark is 200 without having to count the vehicles one by one. The number of vehicles and the number of lots are the same because there is a one to one correspondence between the set of vehicles and the set of lots in the carpark.
By (AP), the desired number of integers is 4 • P28 + 2 • 5 • Pf = 14 • P 2 8 . 3 D There are 10 pupils in a class. (i) ffou; many ways are there to form a 5-member committee for the class? (ii) How many ways are there to form a 5-member committee in which one is the Chairperson, one is the Vice-Chairperson, one is the Secretary and one is the Treasurer? (iii) How many ways are there to form a 5-member committee in which one is the Chairperson, one is the Secretary and one is the Treasurer? Solution (i) This is the same as finding the number of 5-combinations of a 10-element set.
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.