Question

The number of ways in which $20$ letters $a_1,a_2,a_3,\dots ,a_{10},b_1,b_2,b_3,\dots ,b_{10}$ can be arranged in a line so that suffixes of the letters $a$ and also those of $b$ are respectively in ascending order of magnitude is

• A. $\frac{20!}{10!}$
• B. $\frac{20!}{{(10!)}^2}$
• C. $2^{20}$
• D. $20!-10!\cdot 10!$

Answer 1

The correct option is B $\frac{20!}{{(10!)}^2}$

Order of $a^{\prime }$s is fixed that is
$a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9,a_{10}$
These $10$ $a^{\prime }$s can have any $10$ places out of $20$ available places in ${}^{20}{C}_{10}$ ways.
Now, $10$ places are empty where we can insert $b^{\prime }$s in such a way that all $b^{\prime }$s are in ascending order of their suffixes.
Insertion of $b^{\prime }$s can be done in one way only.

So, required number of ways $={}^{20}{C}_{10}\times 1=\frac{20!}{{(10!)}^2}$