Question

If \( {}^9P_3 + 5 \cdot {}^9P_4 = {}^{10}P_r \), then the value of \( 'r' \) is:

• A. \( 4 \)
• B. \( 8 \)
• C. \( 5 \)
• D. \( 7 \)

Hint:

Double-check the formulas and properties of permutations carefully. Ensure the values of \( n \) and \( r \) are correctly substituted.

Answer 1

The Correct Option is C

Step 1: Formula for permutations. The permutation formula is \( {}^nP_k = \frac{n!}{(n - k)!} \).
Step 2: Expand permutation terms. Calculate the values of \( {}^9P_3 \) and \( {}^9P_4 \): \[\n{}^9P_3 = \frac{9!}{(9 - 3)!} = \frac{9!}{6!} = 9 \times 8 \times 7 = 504\n\] \[\n{}^9P_4 = \frac{9!}{(9 - 4)!} = \frac{9!}{5!} = 9 \times 8 \times 7 \times 6 = 3024\n\] The left side of the equation is: \[\n{}^9P_3 + 5 \cdot {}^9P_4 = 504 + 5 \times 3024 = 504 + 15120 = 15624\n\] The right side of the equation is \( {}^{10}P_r = \frac{10!}{(10 - r)!} \). So, \( \frac{10!}{(10 - r)!} = 15624 \).
Step 3: Simplify the left side using properties. The property \( {}^nP_r + r \cdot {}^nP_{r-1} = {}^{(n+1)}P_r \) isn't directly applicable because the coefficient of the second term is 5, not 4. Let's try a different approach: \[\n{}^9P_3 + 5 \cdot {}^9P_4 = {}^9P_3 + 5 \cdot (9-3) \cdot {}^9P_3 = {}^9P_3 (1 + 5 \cdot 6) = 31 \cdot {}^9P_3\n\] This is incorrect. The correct relation is \( {}^nP_r = (n-r+1) {}^nP_{r-1} \), therefore \( {}^9P_4 = (9-4+1) {}^9P_3 = 6 \cdot {}^9P_3 \). Using the definition: \[\n\frac{9!}{6!} + 5 \frac{9!}{5!} = \frac{9!}{6!} (1 + 5 \cdot 6) = 31 \cdot \frac{9!}{6!}\n\] This doesn't seem to simplify to \( {}^{10}P_r \). Consider the identity \( {}^nP_r = {}^nP_{r-1} \cdot (n - r + 1) \). So \( {}^9P_4 = {}^9P_3 \cdot (9 - 4 + 1) = 6 \cdot {}^9P_3 \). Then \( {}^9P_3 + 5 \cdot {}^9P_4 = {}^9P_3 + 30 \cdot {}^9P_3 = 31 \cdot {}^9P_3 = 31 \cdot 504 = 15624 \). Now, we need \( {}^{10}P_r = 15624 \). If \( r = 4 \), \( {}^{10}P_4 = 10 \cdot 9 \cdot 8 \cdot 7 = 5040 \). If \( r = 5 \), \( {}^{10}P_5 = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 = 30240 \). There's an issue. Let's try \( {}^nP_r = {}^nP_{r-1} \frac{n-r+1}{r} \cdot r \). Let's look at the relationship \( {}^nP_r = {}^nP_{r-1} (n - r + 1) \). If \( r = 5 \), \( {}^{10}P_5 = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 = 30240 \). There might be an error in the properties or something is missed. Revisiting: \( {}^9P_3 + 5 \cdot {}^9P_4 = 15624 \). If \( r = 5 \), \( {}^{10}P_5 = 30240 \). Checking the answer choices. If \( r = 5 \), \( {}^{10}P_5 = 30240 \). However, \( {}^9P_3 + 5 \cdot {}^9P_4 = 15624 \neq 30240 \).