An ice hockey team has a total of 17 players consisting of 2 goalies, 6 defender

Question

An ice hockey team has a total of 17 players consisting of 2 goalies, 6 defenders, and 9 forwards. From these 17 players, only 6 players can be on the rink at any time (1 goalie and the remaining 5 who skate on the ice trying to score a goal). The coach has then determined position lines of 2 defenders and 3 forwards. Furthermore, the defenders and forwards each will have assigned positions, that is, left and right for defenders; and left, central, and right for forwards. How many starting lineups can the coach designate?

Explanation / Answer

(a) The one goalie can come in 2 ways.

The 2 defenders can come in 6C2 = 15 ways.

Further, once chosen the two defenders can arrange themselves in 2 ways (left-right or right-left)

=> Total combinations for defenders = 15*2 = 30.

The 3 forwards can come in 9C3 = 84 ways.

Further, once chosen the two defenders can arrange themselves in 3! = 6 ways

=> Total combinations for defenders = 84*6 = 504.

Therefore, the number of starting lineups = 2*30*504 = 30240.

(b) We have seen that the defenders and the forwards had 2 and 6 arrangements due to non-assigned positions.

If the positions were pre-assigned, these arrangements would not have come into picture at all.

The number of starting lineups would be

2*15*84 = 2520.

In other words, the number of assigned starting lineups when players have assigned positions = (1/12) * Number of assigned starting lineups when they don't.

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