Choose people with arrangements

Question

There are 6 participants, 2 male and 4 female, in a singing competition. Three prizes are to be awarded: the 1st, 2nd and 3rd prize. How many possible outcomes are there if

there are no restrictions,
all winners are female,
one of the participants, John, must win a prize.

Solution

Number of ways $= (\binom{6}{3}) \times 3! = 120 ∎$
Number of ways $= (\binom{4}{3}) \times 3! = 24 ∎$
Number of ways $= 3 \times (\binom{5}{2}) \times 2! = 60 ∎$

Skills used

Multiplication principle

Choose and arrange