Question

Find every way that the team can toss the coin during the week and scrimmage first only twice?

*Monday is the first day the Illinois football team tosses a coin for practice. During the 1st 5 days, the team only scrimmages first twice. On possible pattern of the coin is shown.*

Monday - HEADS - Team Scrimmages
Tuesday- HEADS - Team Scrimmages
Wednesday - TAILS - Runs Drills First
Thursday - TAILS - Runs Drills First
Friday - TAILS - Runs Drills First

Answer 1

There are `""_5C_2=10` ways.

Explanation:

We can simplify the question a bit, because "scrimmage first" only occurs when the coin toss is a heads. So, the question reduces to, how many ways can we toss a coin 5 times and get 2 heads?

This is much easier to think about, and much easier to solve. Picture 5 coins in a row, with 2 of them heads and 3 of them tails. This is how many ways we can arrange them:

1. All 5 coins can be arranged in `5!` (or 120) orders.
2. Since we don't care about the order of the two heads, we divide by the `2!` (2) permutations those two heads make.
3. Again we don't care about the order of the three tails, so we divide by the `3!` (6) permutations the three tails make.

Our final number is `(5!)/(2!xx3!)`, which equals `120/(2xx6),` or 10 ways to arrange two heads and three tails.

(Quick example: with 5 coins, you could have `H_1H_2 T T T,` but we count that the same as `H_2H_1T T T.` They may be different orders of all 5 coins, but the result is still "2 heads followed by 3 tails".)

This is a common probability concept called combinations. It's how we count the number of ways to select `k` things (like 2 heads) from a group of `n` things (5 coins), if we don't care about the order in which the similar things (2 heads, 3 tails) are chosen.

This value is written as `""_nC_k` or `((n),(k))`, and is spoken as "n choose k". It is equal to `(n!)/(k!(n-k)!).`

The 10 combinations of 2 H and 3 T are:

`H H T T T" "T H T H T`
`H T H T T" "T H T T H`
`H T T H T" "T T H H T`
`H T T T H" "T T H T H`
`T H H T T" "T T T H H`