A coach is filling out the starting lineup for his indoor soccer team.
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A coach is filling out the starting lineup for his indoor soccer team. There are 10 boys on the team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defence, 2 in midfield, and 1 forward. Only 2 of the boys can play goalkeeper, and they cannot play any other positions. The other boys can each play any of the other positions.
How many different groupings are possible?
Show answer & explanation
Choose 1 of the 2 goalkeeper-only players for goalkeeper, then from the remaining 8 boys choose and assign 2 defenders, 2 midfielders, and 1 forward.
Step-by-step Derivation:
Step 1: Choose the goalkeeper from the 2 boys who can play goalkeeper: 2 ways.
Step 2: The goalkeeper cannot play elsewhere, so the remaining 8 boys are available for the 5 outfield positions.
Step 3: Choose 2 defenders from 8 boys: C(8,2) = 28 ways.
Step 4: Choose 2 midfielders from the remaining 6 boys: C(6,2) = 15 ways.
Step 5: Choose 1 forward from the remaining 4 boys: C(4,1) = 4 ways.
Step 6: Multiply the independent choices: 2 × 28 × 15 × 4 = 3360.
So the number of possible different groupings is 3360.