First slide

Permutations

Question

$The value of (2 \cdot {}^{1}P_{0} - 3 \cdot {}^{2}P_{1} + 4 \cdot {}^{3}P_{2} - \dots \dots \dots . upto 51^{th} term) + (1! - 2! + 3! - \dots \dots up to 51^{th term}) is$ equal to :

Moderate

A

$1 - 51 (51)!$
B

$1 + (52)!$
C

1
D

$1 + (51)!$

Solution

$\begin{array}{l} S = (2^{1} P_{0} - 3^{2} P_{1} + 4^{3} P_{2} + \dots \dots 51 tems) + (1! - 2! + 3! - 4! + \dots \dots 51 terms) \\ = (2.1 - 3.2! + 4.3! + . . . . . . . . - 51 (50)! + 52 (51)!) + 1! - 2! + 3! - 4! + \dots . . . . - 50! + 51!) \\ = 2! - 3! + 4! - 5! + \dots \dots - 51! + 52! + (1! - 2! + 3! - 4! + \dots . . + 51!) = 1 + 52! \end{array}$

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