Question

Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to the number of heads is?

• A. $20$
• B. $9$
• C. $120$
• D. $40$

Answer 1

The correct option is A

$20$

Explanation for the correct answer:

Finding the number of ways in which the number of tails is equal to heads:

Let's arrange six identical coins in a row.

For the number of heads and tails to be equal, the arrangement is as follows

$\left\{H,H,H,T,T,T\right\}$

Hence, the arrangement is $3$ consecutive heads and $3$ consecutive tails.

The total number of arrangements of six coins $=6!$

Therefore, the required number of ways$={}^{6}C_3$

$$\begin{gathered} =\frac{6!}{3!.3!} \\ =\frac{6\times 5\times 4\times 3\times 2\times 1}{3\times 2\times 1\times 3\times 2\times 1} \\ =20 \end{gathered}$$

The number of ways in which the number of tails is equal to the number of heads is $20$.

Therefore, option (A) is the correct answer.