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Question: Ram and Shyam select two numbers from 1 to n. If the probability that Shyam selects a number is less...

Ram and Shyam select two numbers from 1 to n. If the probability that Shyam selects a number is less than the number selected by Ram is 63128\dfrac{{63}}{{128}}, then
A) n is odd
B) n is a perfect square
C) n is a cube
D) None of these

Explanation

Solution

The number of ways of selecting two numbers from n numbers is nC2{}^n{C_2}.
So, the probability that probability that Shyam selects a number is less than the number selected by Ram is given by nC2n×n\dfrac{{{}^n{C_2}}}{{n \times n}} .
Now, solve nC2n×n\dfrac{{{}^n{C_2}}}{{n \times n}} and compare it with 63128\dfrac{{63}}{{128}}.
On getting the value of n, look at the number if it is odd, perfect square or a cube.

Complete step by step solution:
It is given that Ram and Shyam select two numbers from 1 to n.
So, the number of ways of selecting two numbers from n numbers is nC2{}^n{C_2} .
Ram and Shyam both have the same number of choices i.e. n. So, the number of possibilities of Ram and Shyam selecting a number is n×nn \times n .
Now, the probability that probability that Shyam selects a number is less than the number selected by Ram is given by nC2n×n\dfrac{{{}^n{C_2}}}{{n \times n}} .
nC2n×n=n!2!(n−2)!n2 =n(n−1)(n−2)!2n2(n−2)! =n−12n  \dfrac{{{}^n{C_2}}}{{n \times n}} = \dfrac{{\dfrac{{n!}}{{2!\left( {n - 2} \right)!}}}}{{{n^2}}} \\\ = \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)!}}{{2{n^2}\left( {n - 2} \right)!}} \\\ = \dfrac{{n - 1}}{{2n}} \\\
It is given that the probability that Shyam selects a number is less than the number selected by Ram is 63128\dfrac{{63}}{{128}} .
Thus, n−12n=63128\dfrac{{n - 1}}{{2n}} = \dfrac{{63}}{{128}}
∴128(n−1)=2×63×n ∴128n−128=126n ∴2n=128 ∴n=64  \therefore 128\left( {n - 1} \right) = 2 \times 63 \times n \\\ \therefore 128n - 128 = 126n \\\ \therefore 2n = 128 \\\ \therefore n = 64 \\\
Thus, we get n as 64, which is a square of 8.

So, option (B) is correct.

Note:
Here, the condition that Shyam selects a number is less than the number selected by Ram, does not affect the number of choosing two numbers from n numbers as we have directly counted the number of selecting two numbers from n numbers and then we assign the higher number to Ram and smaller number to Shyam.