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

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

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

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N n choices n-\ choices 1 n-(r-2) n-0-l) choices choices n—2 choices ... 1 rth Counting 20 We wish to choose r elements from {1, 2 , . . ,n} to fill the r spaces, where the ordering of elements matters. There are n choices for the 1st space. After fixing one in the 1st space, there are n — 1 choices remaining for the 2nd space. After fixing one in the 2nd space, there are n — 2 choices left for the 3rd space, and so on. After fixing one in the (r — l)th space, there are n — (r — 1) choices left for the rth space.

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.

4 {{A,B},{C,D}}, {{A,C},{B,D}}, {{A,D},{B,C}}. In how many ways can 10 people be paired off? If n points on the circumference of a circle are joined by straight lines in all possible ways and no three of these lines meet at a single point inside the circle, find (i) the number of triangles formed with all vertices lying inside the circle; (ii) the number of triangles formed with exactly two vertices inside the circle; (iii) the number of triangles formed with exactly one vertex inside the circle; (iv) the total number of triangles formed.

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