Question

If $m$ denotes the number of $5$ digit numbers if each successive digits are in their descending order of magnitude and $n$ is the corresponding figure, when the digits are in their ascending order of magnitude then $(m-n)$ has the value.

• A. ${{}^{24}C}_4$
• B. ${{}^{9}C}_5$
• C. ${{}^{10}C}_5$
• D. ${{}^{9}C}_3$

Answer 1

Answer: (B)

${{}^{9}C}_5$

For descending order : to form a five digit no we need to choose $5$ numbers out of $10$ numbers, which is ${{}^{10}C}_5$. Now you don't need to arrange them, there is only one way by which $5$ different numbers can be arranged in only one way.

But for ascending order we have only $9$ options to choose from, as including $0$ and arranging it in ascending order will put on the front and with zero on front the number will be a $4$ digit number and not $5$ (For ex : if you choose$0,1,2,3,4$ the ascending order will be $01234$ but this is a four digit number and not $5$) So for this case we have ${{}^{9}C}_4$ cases.

So $m-n$ gives, ${{}^{10}C}_5-{{}^{9}C}_5={{}^{9}C}_4={{}^{9}C}_5$

Hence, $\left(B\right)$