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 value.
A. ${}^{24}{C_4}$
B. ${}^9{C_5}$
C. ${}^{10}{C_5}$
D. ${}^9{C_3}$

Answer 1

Hint- In order to solve this first separately use the concept of permutation and combination to find the figure for descending order of the numbers and ascending order of numbers. Finally subtract the results and use the formula for the subtraction of combination of terms to get the correct answer amongst the given options.

Complete answer:
For descending order
To form a five digit number we need to choose 5 numbers out of 10 numbers
This can be done in ​${}^{10}{C_5}$ ways.
Now we do not need to arrange them, as there is only one way by which 5 different numbers can be arranged
$\Rightarrow m = {}^{10}{C_5}$
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 no will be a 4 digit no and not 5
So for this case we have ${}^9{C_4}$​ cases.
$\Rightarrow n = {}^9{C_4} \\\ \Rightarrow m - n = {}^{10}{C_5} - {}^9{C_4} = {}^9{C_5}{\text{ }}\left[ {\because {}^m{C_n} - {}^{m - 1}{C_{n - 1}} = {}^{m - 1}{C_n}} \right] \\\$
Thus,
$\Rightarrow m - n = {}^9{C_5}$

Hence correct option is “B”

Note- In order to solve these types of problems, first of all remember all the formulas of combination. Also remember the properties of combinations. The next step is to read the statement carefully and write the conditions given in the questions and solve accordingly. You must have a good command on algebra to and algebraic identities.