Question

How many ways can you pick a president, vice-president, and secretary from a group of six boys and five girls if
a) there are no restrictions?
b) there must be at least one boy chosen?
c) there must be only one girl chosen?

Answer 1

There are 3 positions. There are 6 boys and 5 girls. Total of 11

a) no restrictions

This is a permutation ;

_nP_{r =}$\frac {n!} {(n-r)!}$

=​$\frac {11!}{(11-3)!}$

=990 ways

b) at least one boy chosen.

This is where 1, 2 or 3 boys can be selected.

We subtract when no boy(all girls) is selected from when no restriction is put in place.

₁₁P₃-₅P₃

=$\frac {11!}{(11-3)!}-\frac {5!}{(5-3)!}​$

=990- 60

=930 ways

c) only one girl chosen

This means that 2 boys are chosen

And there are 3 positions.

₅P₁*₆P₂*3

=$\frac {5!}{(5-1)!}*\frac {6!}{(6-2)!}*3$

=450 ways