Question

If \( _nP_r = 480 \) and \( _nC_r = 20 \), then the value of \( r \) is equal to:

• A. 2
• B. 3
• C. 4
• D. 5
• E. 6

Hint:

To solve for \( r \), use the relationship between permutations and combinations: \( _nP_r = r! \times _nC_r \).

Answer 1

The Correct Option is C

Using the formulas for permutations and combinations: \[ _nP_r = \frac{n!}{(n-r)!} = 480 \] \[ _nC_r = \frac{n!}{r!(n-r)!} = 20 \] Dividing the first equation by the second: \[ \frac{_nP_r}{_nC_r} = \frac{\frac{n!}{(n-r)!}}{\frac{n!}{r!(n-r)!}} = r! \] \[ \frac{480}{20} = r! \implies r! = 24 \] Since \( 4! = 24 \), we conclude: \[ r = 4 \]