Advertisements
Advertisements
Question
Prove that:1 . P (1, 1) + 2 . P (2, 2) + 3 . P (3, 3) + ... + n . P (n, n) = P (n + 1, n + 1) − 1.
Advertisements
Solution
1.P (1, 1) + 2. P (2, 2) + 3. P (3, 3) + ... + n . P (n, n) = P (n + 1, n + 1) − 1
P (n,n) = n!
1.1! + 2.2! + 3.3! ......+ n.n! = (n+1)! − 1
LHS = 1.1! + 2.2! + 3.3! ......+ n.n!
\[= \sum^n_{r = 1} r . r!\]
\[ = \sum^n_{r = 1} \left[ \left( r + 1 \right) - 1 \right] r!\]
\[ = \sum^n_{r = 1} \left[ \left( r + 1 \right) r! - r! \right]\]
\[ = \sum^n_{r = 1} {(r + 1)! - r!} \]
\[ = \left( 2! - 1! \right) + \left( 3! - 2! \right) + . . . \left[ \left( n + 1 \right)! - n! \right]\]
\[ = \left[ \left( n + 1 \right)! - 1! \right] \]
\[ = \left[ \left( n + 1 \right)! - 1 \right] = \text{RHS}\]
\[ \text{Hence, proved} .\]
APPEARS IN
RELATED QUESTIONS
Convert the following products into factorials:
(n + 1) (n + 2) (n + 3) ... (2n)
Convert the following products into factorials:
1 · 3 · 5 · 7 · 9 ... (2n − 1)
Prove that:
Prove that:
If P (n − 1, 3) : P (n, 4) = 1 : 9, find n.
In how many ways can five children stand in a queue?
How many three-digit numbers are there, with distinct digits, with each digit odd?
How many words, with or without meaning, can be formed by using all the letters of the word 'DELHI', using each letter exactly once?
How many three-digit numbers are there, with no digit repeated?
How many 3-digit even number can be made using the digits 1, 2, 3, 4, 5, 6, 7, if no digits is repeated?
In how many ways can the letters of the word 'FAILURE' be arranged so that the consonants may occupy only odd positions?
How many words can be formed from the letters of the word 'SUNDAY'? How many of these begin with D?
How many words can be formed out of the letters of the word, 'ORIENTAL', so that the vowels always occupy the odd places?
How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:
the letter G always occupies the first place?
How many permutations can be formed by the letters of the word, 'VOWELS', when
each word begins with O and ends with L?
How many permutations can be formed by the letters of the word, 'VOWELS', when
all consonants come together?
How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?
m men and n women are to be seated in a row so that no two women sit together. if m > n then show that the number of ways in which they can be seated as\[\frac{m! (m + 1)!}{(m - n + 1) !}\]
How many words (with or without dictionary meaning) can be made from the letters in the word MONDAY, assuming that no letter is repeated, if all letters are used at a time.
How many words (with or without dictionary meaning) can be made from the letters in the word MONDAY, assuming that no letter is repeated, if all letters are used but first is vowel.
Find the number of words formed by permuting all the letters of the following words:
ARRANGE
Find the number of words formed by permuting all the letters of the following words:
INDIA
In how many ways can the letters of the word 'ALGEBRA' be arranged without changing the relative order of the vowels and consonants?
Find the total number of arrangements of the letters in the expression a3 b2 c4 when written at full length.
How many different numbers, greater than 50000 can be formed with the digits 0, 1, 1, 5, 9.
Find the number of numbers, greater than a million, that can be formed with the digits 2, 3, 0, 3, 4, 2, 3.
A biologist studying the genetic code is interested to know the number of possible arrangements of 12 molecules in a chain. The chain contains 4 different molecules represented by the initials A (for Adenine), C (for Cytosine), G (for Guanine) and T (for Thymine) and 3 molecules of each kind. How many different such arrangements are possible?
In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?
Find the total number of permutations of the letters of the word 'INSTITUTE'.
Prove that: 4nC2n : 2nCn = [1 · 3 · 5 ... (4n − 1)] : [1 · 3 · 5 ... (2n − 1)]2.
There are 10 persons named\[P_1 , P_2 , P_3 , . . . . , P_{10}\]
Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement P1 must occur whereas P4 and P5 do not occur. Find the number of such possible arrangements.
How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used at a time
How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used but first letter is a vowel?
Find the number of permutations of n different things taken r at a time such that two specified things occur together?
Write the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants.
