NIMCET 2021 — Mathematics PYQ

Solution

To minimize the expression $n!(21-n)!$, we can relate it to the formula for binomial coefficients.

Step 1: Relate the expression to ${}^{21}{C}_n$.

The formula for ${}^n{C}_r$ is:

In this problem, let $N=21$ and $r=n$. Then:

Step 2: Understand the relationship between the expression and the coefficient.

From the formula above, we can write:

Since $21!$ is a constant value, the expression $n!(21-n)!$ will be minimum when the denominator ${}^{21}{C}_n$ is maximum.

Step 3: Find the maximum value of ${}^{21}{C}_n$.

The binomial coefficient ${}^N{C}_n$ reaches its maximum value at the middle term(s).

• If $N$ is even, the maximum value occurs at $n=\frac{N}{2}$.

• If $N$ is odd, the maximum value occurs at $n=\frac{N-1}{2}$ and $n=\frac{N+1}{2}$.

Step 4: Calculate the value of $n$.

Here, $N=21$, which is an odd number. Therefore, the maximum values of ${}^{21}{C}_n$ occur at:

Conclusion:

The value of $n!(21-n)!$ is minimized when $n$ is either $10$ or $11$.

Final Answer:

The values of $n$ are $10$ or $11$.