Question

3. A sailing club has nine human members (1,...,9) and four sailboats (A,B,C,D). (a) How many distinct two-member crews can be formed? (b) In how many distinct ways can all four sailboats be assigned two-member crews? Note: A correct formula in terms of binomial coefficients is a sufficient answer.

Answer 1

(a)

There are 9 people in all

2 are to selected so there will be ⁹C₂ ways which add up to 36 ways in total.

A crude method is :

1,2   

1,3 2,3

1,4 2,4 3,4

1,5 2,5 3,5 4,5

1,6 2,6 3,6 4,6 5,6

1,7 2,7 3,7 4,7 5,7 6,7

1,8 2,8 3,8 4,8 5,8 6,8 7,8

1,9 2,9 3,9 4,9 5,9 6,9 7,9 8,9

Note : 1,1 or any combination won't come as else it won't be 2 crew and 2,1 or etc won't come as else it won't be distinct i.e. (1,2)=(2,1)

(b) We now need to have distinct people in team and (1,2) and (2,3) cant take two boats as 2 is getting repeated.

So, we need to select 8 distinct out of 9 people . so there are 9 ways of doing this. Now we make teams of 2 members so 4 teams. So , there are 4 *4 =16 ways of giving the boats.