Question

The number of different ways in which $5$ 'dashes' and $8$ 'dots' can be arranged, using only $7$ of these 'dashes' and 'dots', is

• A. $1287$
• B. $119$
• C. $120$
• D. $1235520$

Answer 1

The correct option is C $120$

We know all 'dashes' are identical and all 'dots' are identical.

So total ways of selecting and permuting 'dashes' and 'dots' are:

$\left(i\right)7$ dots $\Rightarrow \frac{7!}{7!}=1$

$\left(ii\right)6$ dots $1$ dashes $\Rightarrow \frac{7!}{6!1!}=7$

$\left(iii\right)5$ dots $2$ dashes $\Rightarrow \frac{7!}{5!2!}=21$

$\left(iv\right)4$ dots $3$ dashes $\Rightarrow \frac{7!}{4!3!}=35$

$\left(v\right)3$ dots $4$ dashes $\Rightarrow \frac{7!}{3!4!}=35$

$\left(vi\right)2$ dots $5$ dashes $\Rightarrow \frac{7!}{2!5!}=21$

Total ways of arranging is $1+7+21+35+35+21=120$

Hence, $\left(C\right)$