=\( \dfrac{n!}{0!}\)
= n! [Since, 0! = 1]
Consider LHS:
= 1. P(1, 1) + 2. P(2, 2) + 3. P(3, 3) + … + n . P(n, n)
= 1.1! + 2.2! + 3.3! +………+ n.n! [Since, P(n, n) = n!]
= (2! – 1!) + (3! – 2!) + (4! – 3!) + ……… + (n! – (n – 1)!) + ((n+1)! – n!)
= 2! – 1! + 3! – 2! + 4! – 3! + ……… + n! – (n – 1)! + (n+1)! – n!
= (n + 1)! – 1!
= (n + 1)! – 1 [Since, P (n, n) = n!]
= P(n+1, n+1) – 1
= RHS
Hence Proved.
Answered by Sakshi | 1 year agoHow many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?
Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.
How many words, with or without meaning can be formed from the letters of the word ‘MONDAY’, assuming that no letter is repeated, if
(i) 4 letters are used at a time
(ii) all letters are used at a time
(iii) all letters are used but first letter is a vowel ?
There are 10 persons named P1, P2, P3 …, P10. Out of 10 persons, 5 persons are to be arranged in a line such that is each arrangement P1 must occur whereas P4 and P5 do not occur. Find the number of such possible arrangements.
How many different words, each containing 2 vowels and 3 consonants can be formed with 5 vowels and 17 consonants?