JEE PYQ: Permutation And Combination Question 17
Question 17 - 2020 (03 Sep Shift 1)
The value of $(2 \cdot {}^1P_0 - 3 \cdot {}^2P_1 + 4 \cdot {}^3P_2 - \cdots$ up to $51^{\text{th}}$ term) $+ (1! - 2! + 3! - \cdots$ up to $51^{\text{th}}$ term) is equal to:
(1) $1 - 51(51)!$
(2) $1 + (51)!$
(3) $1 + (52)!$
(4) $1$
Show Answer
Answer: (3)
Solution
$(r+1) \cdot {}^rP_{r-1} = (r+1) \cdot \frac{r!}{1!} = (r+1)!$. So first sum $= 2! - 3! + 4! - \cdots + 52!$. Second sum $= 1! - 2! + 3! - \cdots + 51!$. Adding: $= [2! - 3! + 4! - \cdots + 52!] + [1! - 2! + 3! - \cdots + 51!] = 52! + 1! = 1 + 52!$.
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