Numerical Applications Test 51

Work efficiency of pump A in emptying pool = $\dfrac{1}{4}$ per hour

Work efficiency of pump B in emptying pool = $\dfrac{1}{6}$ per hour

$\therefore$ work done by A and B together = $\dfrac{1}{4} + \dfrac{1}{6} = \dfrac{10}{24}$per hour

Hence time taken to fill the tank  $100$% = $\dfrac{24}{10}$ hours

$\therefore$ time taken to fill $50$% = $\dfrac{12\times 50}{5\times100}$  = $1.2$ hours

Since, $1 hour = 60 minutes$

$\therefore 1.2$ hours= $1.2\times 60 = 72$ minutes

$72$ minutes = $(60 +12)$ minutes = $1 hour$  $12 minutes$

So, the answer is option A

${\textbf{Step 1: Find m}}$

                ${\text{For m,}}$

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

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

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

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

${\textbf{Step 2: Find n}}$

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

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

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

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

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

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

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

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

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

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