Question

In how many ways can the season end with 8 wins, 4 losses, and 2 tie is a college football team plays 14 games?

Answer 1

Number of Combinations

Let `C(n,r)` denote the number of combinations of `n` items chosen `r` items at a time.

`C(n,r)={n!}/{(n-r)!r!}={n cdot(n-1)cdot(n-2)cdotcdotscdot(n-r+1)}/{r!}`

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The number of ways to tie 2 out of 14 games can be found by

`C(14,2)={14cdot13}/{2cdot1}=91`.

The number of ways to lose 4 out of 12 remaining games can be found by

`C(12,4)={12cdot11cdot10cdot9}/{4cdot3cdot2cdot1}=495`.

The number of ways to win 8 out of 8 remaining games can be found by

`C(8,8)={8!}/{8!}=1`

Hence, the total number of ways to have 8-4-2 record is

`91cdot495cdot1=45045`.

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I hope that this was helpful.