Question

There are $10$ red and $5$ yellow roses of different sizes. If $x$ is the number of garlands that can be formed with all these flowers so that no two yellow roses come together and $y$ is the number of garlands formed with all these flowers so that all the red roses coming together, then $\frac{2(x-y)}{10!}=$

• A. $\frac{9!}{5!}-5!$
• B. $(11{)}^2.(4!)$
• C. $10!-6!$
• D. $6!\times (5!-2)$

Answer 1

Answer: (A)

We have, $10$ Red and $5$ Yellow roses of different sizes.
Now, $x=$ number of garlands that can be formed with all these flowers
so that no two yellow roses come together
$=\frac{1}{2}(10-1)!\times {}^{10}{P}_5$
$=\frac{1}{2}\times 9!\times \frac{10!}{5!}=x=\frac{10!}{2}\left[\frac{9!}{5!}\right]$
$\Rightarrow \frac{2x}{10!}=\frac{9!}{5!}$
and $y=$ number of garlands formed with all these flowers so that all the red roses coming together
$=\frac{1}{2}(6-1)!\times 10!=\frac{1}{2}\times 5!\times 10!$
$\Rightarrow \frac{2y}{10!}=5!$
$\therefore \frac{2(x-y)}{10!}=\frac{9!}{5!}-5!$