Permutations and Combinations Test 48

$$  {\textbf{Step 1: Find m}} $$

                $$  {\text{For m,}} $$

                $$  {\text{First we select any 5 digits from 0,1,2,}}...{\text{,9}} $$

                $$  {\text{Number of ways = }}{}^{10}{{\text{C}}_5} $$

                $$  {\text{Now after selection there is only 1 way to arrange these selected digits, i}}{\text{.e}}{\text{., in descending order}}{\text{.}} $$

                $$  {\text{Therefore m = }}{}^{10}{{\text{C}}_5}\times{\text{  1 = }}{}^{10}{{\text{C}}_5} $$

$$  {\textbf{Step 2: Find n}} $$

                $$  {\text{For n,First we select any 5 digits from 1,2,}}...{\text{,9}} $$

                $$  {\text{We can't select zero as  first digit because then the number won't be a 5 - digit number}}{\text{.}} $$

                $$  {\text{Therefore number of ways  = }}{}^9{{\text{C}}_5} $$

                $$   \Rightarrow {\text{n = }}{}^9{{\text{C}}_5}\times{\text{  1 = }}{}^9{{\text{C}}_5} $$

                $$   \Rightarrow {\text{m - n = }}{}^{10}{{\text{C}}_5}{\text{ - }}{}^9{{\text{C}}_5} $$ 

                $$  {\text{We know that,}}{}^n{{\text{C}}_r}{\text{ + }}{}^n{{\text{C}}_{r - 1}}{\text{ = }}{}^{n + 1}{{\text{C}}_r} $$

                $$   \Rightarrow {}^{n + 1}{{\text{C}}_r}{\text{ - }}{}^n{{\text{C}}_r}{\text{ = }}{}^n{{\text{C}}_{r - 1}} $$

                $$   \Rightarrow {}^{10}{{\text{C}}_5}{\text{ - }}{}^9{{\text{C}}_5}{\text{ = }}{}^9{{\text{C}}_4} $$

                $$  {\text{Hence, m - n = }}{}^9{{\text{C}}_4} $$

$$  {\textbf{Hence, the correct answer is option A}} $$

Each place of a ten digit number can be fixed by any of the two digits. So, the number of ways to form a ten digit number is $${2^{10}}$$.