Question

A coin is tossed $20$ times and the heads and tails sequence is recorded.

Find among all the possible sequence of heads and tails, how many have exactly seven heads?

Answer 1

Find the possible sequence of heads and tails that have exactly seven heads:

The coin is tossed $20$ times, we need to find number of possible sequence of heads and tails that have exactly seven heads.

The formula of finding number of possible arrangements of selecting $r$ object from the set of $n$ without repetition is given by:

${}^{n}C_r=\frac{n!}{r!\left(n-r\right)!}$

$\therefore$The number of possible sequence of heads and tails that have exactly $7$ heads in the $20$ tosses of a coin is:

$\begin{array}{rcl}{{}^{20}C}_7& =& \frac{20!}{7!\left(20-7\right)!}\\ & =& \frac{20!}{7!13!!}\\ & =& 77,520\end{array}$

Therefore, the possible sequence of heads and tails that have exactly seven heads if the coin is tossed $20$ times is $77,520$.