Question

Two natural numbers are chosen at random from the first one hundred natural numbers. The probability that the product of the chosen numbers is a multiple of 7 is.

• A. $\frac{{14}_{C_1}\times {13}_{C_1}}{{100}_{C_2}}$
• B. $\frac{{14}_{C_2}}{{100}_{C_2}}+\frac{{14}_{C_1}{13}_{C_1}}{{100}_{C_2}}$
• C. $\frac{{14}_{C_1}{86}_{C_1}}{{100}_{C_2}}+\frac{{14}_{C_2}}{{100}_{C_2}}$
• D. $\frac{{14}_{C_2}}{{100}_{C_2}}+\frac{7_{C_1}{14}_{C_1}}{{100}_{C_2}}$

Answer 1

The correct option is C

$\frac{{14}_{C_1}{86}_{C_1}}{{100}_{C_2}}+\frac{{14}_{C_2}}{{100}_{C_2}}$

Out of the 100 natural numbers, 14 numbers are multiples of 7.
These exists two cases.

Case 1: The product is a multiple of 7 if one of the number is in the form of 7k.
$\Rightarrow \frac{{14}_{C_1}\times {86}_{C_1}}{{100}_{C_2}}$

Case 2: The product is a multiple of 7 if both the numbers are in the form of 7k.
$\Rightarrow \frac{{14}_{C_2}}{{100}_{C_2}}$

$\therefore \text{Probability}=\frac{{14}_{C_1}{86}_{C_1}}{{100}_{C_2}}+\frac{{14}_{C_2}}{{100}_{C_2}}$