Question

If \(^{10}P_r = 604800\) and \(^{10}C_r = 120\), then \(r\) is:

• A. 14
• B. 7
• C. 8
• D. 16

Hint:

Remember that \(nP_r = nC_r \cdot r!\); use this relation to find unknown \(r\) when \(nP_r\) and \(nC_r\) are given.

Answer 1

The Correct Option is B

Step 1: Relate nPr and nCr directly.
Since \(^nP_r = {}^nC_r \times r!\), we can solve for \(r!\) without expanding \(10!\) at all.

Step 2: Substitute the given values.
\[ r! = \frac{{}^{10}P_r}{{}^{10}C_r} = \frac{604800}{120}=5040 \]

Step 3: Identify which factorial equals 5040.
\(7!=7\times6\times5\times4\times3\times2\times1=5040\), so \(r=7\).

Step 4: Conclusion.
\[ \boxed{7} \]