JAMIA 2024 — Mathematics PYQ

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Question

JAMIA 2024 — Mathematics PYQ

JAMIA | Mathematics | 2024

$If^{n} P_{3} = 4 \times^{n} P_{2}, find n .$

Choose the correct answer:

Explanation

Step 1: Use the Permutation Formula

The formula for permutations is given by:

{}^{n}P_{r} = \frac{n!}{(n-r)!}

Applying this to both sides of the equation:

\frac{n!}{(n-3)!} = 4 \times \frac{n!}{(n-2)!}

Step 2: Simplify the Factorials

We can expand the factorials to simplify the expression:

$\frac{n!}{(n-3)!} = n(n-1)(n-2)$
$\frac{n!}{(n-2)!} = n(n-1)$

Now, substitute these back into the equation:

n(n-1)(n-2) = 4 \times n(n-1)

Step 3: Solve for $n$

Since $n \geq 3$ (for ${}^{n}P_{3}$ to be defined), we know that $n \neq 0$ and $n \neq 1$ . Therefore, we can divide both sides by $n(n-1)$ :

n - 2 = 4

Adding $2$ to both sides:

n = 4 + 2

n = 6

Final Answer

The value of $n$ is:

n = 6