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
$=\frac{1}{2}(10-1) ! \times{ }^{10} P_{5}$
$=\frac{1}{2} \times 9 ! \times \frac{10 !}{5 !}=x=\frac{10 !}{2}\left[\frac{9 !}{5 !}\right]$
$\Rightarrow \frac{2 x}{10 !}=\frac{9 !}{5 !}$
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{2 y}{10 !}=5 ! $
$\therefore \frac{2(x-y)}{10 !}=\frac{9 !}{5 !}-5 !$