JAMIA 2024 — Mathematics PYQ

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Question

JAMIA 2024 — Mathematics PYQ

JAMIA | Mathematics | 2024

$^{n} P_{r} =^{n} C_{r} x$

Choose the correct answer:

Explanation

Step 1: Write the Formulas

The formula for Permutations is:

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

The formula for Combinations is:

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

Step 2: Relationship between $P$ and $C$

We can rewrite the combination formula to show its relationship with the permutation formula:

{}^{n}C_{r} = \frac{1}{r!} \times \left( \frac{n!}{(n-r)!} \right)

{}^{n}C_{r} = \frac{{}^{n}P_{r}}{r!}

Step 3: Solve for $x$

The given equation is:

{}^{n}P_{r} = {}^{n}C_{r} \cdot x

Substitute the relationship ${}^{n}P_{r} = r! \cdot {}^{n}C_{r}$ into the equation:

r! \cdot {}^{n}C_{r} = {}^{n}C_{r} \cdot x

Dividing both sides by ${}^{n}C_{r}$ :

x = r!

Final Answer

The value of $x$ is:

r!