# Coin Flip: “Exactly” and “At Most”

A coin is flipped $10$ times. How many outcomes have exactly three heads? How many outcomes have at most three heads?

• I suspect that we need to put slots between $10-3=7$ tails so that we end up choosing $3$ from these $7$ empty slots: $_T_T_T_T_T_T_T_$ – Camron Ignazio Jul 23 '13 at 3:05
• Does $HHHTTTTTTT$ fit in that pattern? – Daniel Fischer Jul 23 '13 at 3:07
• Do you know about binomial coefficients? For example, do you know what ${10\choose 3}$ means? – Potato Jul 23 '13 at 3:08
• The number of ways $3$ things can be chosen from $10$ things. – Camron Ignazio Jul 23 '13 at 3:09

Imagine writing down a sequence of length $10$, made up of the letters H and/or T, to indicate what happened on your tosses.
Exactly $3$ heads happened precisely if the sequence has exactly $3$ H (and therefore $7$ T).
There are $\binom{10}{3}$ ways to choose where the $3$ H will go, so there are $\binom{10}{3}$ such sequences.
For at most $3$ heads, do the same sort of thing for $0$ H, $1$ H, $2$ H, and $3$ H, and add up.