NIMCET 2015 — Mathematics PYQ

The formula for permutation ${}^n{P}_r$ is given by:

Alternatively, it can be written as the product of $r$ consecutive decreasing integers starting from $n$:

2. Set up the Equation

Given the equation:

Using the expansion method for both sides:

• ${}^n{P}_2=n(n-1)$

• ${}^n{P}_4=n(n-1)(n-2)(n-3)$

Substituting these into the equation:

3. Simplify the Equation

For ${}^n{P}_4$ to be defined, $n$ must be $\ge 4$. Therefore, $n$ and $(n-1)$ cannot be zero. We can safely divide both sides by $n(n-1)$:

4. Solve for $n$

Expand the right side:

Rearrange into a quadratic equation:

Factorize the quadratic:

This gives two possible values for $n$:

1. $n=9$

2. $n=-4$

Conclusion

Since the number of items $n$ cannot be negative, we discard $n=-4$.

Thus, $n=9$.

Correct Option: (c)