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Permutation and Combination - Quant/Math - CAT 2016

Question 4 the day: May 9, 2002

Last few CATs have had a lot of emphasis on problems from Number Systems, Inequalities and functions (CAT 2001 was an exception when it comes to functions) in the quant section. Many of these questions were based on math concepts known to us but that were put in a slightly different way.

Here is one such typical quant problem which appears in CAT and other MBA entrance exams which looks like Greek and Latin at the first instance but is actually a pretty easy one.

Question:
What is the value of 1*1! + 2*2! + 3!*3! + ............ n*n!,
where n! means n factorial or n(n-1)(n-2)...1

(1) n(n-1)(n-1)!
(2) (n+1)!/(n(n-1))
(3) (n+1)! - n!
(4) (n + 1)! - 1!

Solution:

1*1! = (2 -1)*1! = 2*1! - 1*1! = 2! - 1!
2*2! = (3 - 1)*2! = 3*2! - 2! = 3! - 2!
3*3! = (4 - 1)*3! = 4*3! - 3! = 4! - 3!
..
..
..
n*n! = (n+1 - 1)*n! = (n+1)(n!) - n! = (n+1)! - n!

Summing up all these terms, we get (n+1)! - 1!

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