Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Convert the following products into factorials:
5 · 6 · 7 · 8 · 9 · 10
If (n + 2)! = 60 [(n − 1)!], find n.
If P (5, r) = P (6, r − 1), find r ?
If nP4 = 360, find the value of n.
If P(11, r) = P (12, r − 1) find r.
From among the 36 teachers in a school, one principal and one vice-principal are to be appointed. In how many ways can this be done?
Four books, one each in Chemistry, Physics, Biology and Mathematics, are to be arranged in a shelf. In how many ways can this be done?
There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible, correct or incorrect, answers are there to this question?
In how many ways can the letters of the word 'FAILURE' be arranged so that the consonants may occupy only odd positions?
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 vowels come together?
How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?
Find the number of words formed by permuting all the letters of the following words:
INTERMEDIATE
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:
PAKISTAN
Find the number of words formed by permuting all the letters of the following words:
RUSSIA
Find the total number of arrangements of the letters in the expression a3 b2 c4 when written at full length.
How many different signals can be made from 4 red, 2 white and 3 green flags by arranging all of them vertically on a flagstaff?
In how many ways can the letters of the word 'ARRANGE' be arranged so that the two R's are never together?
How many different numbers, greater than 50000 can be formed with the digits 0, 1, 1, 5, 9.
How many words can be formed from the letters of the word 'SERIES' which start with S and end with S?
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 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable?
In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?
If the letters of the word 'LATE' be permuted and the words so formed be arranged as in a dictionary, find the rank of the word LATE.
If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary, find the rank of the word 'MOTHER'.
Find the total number of ways in which six ‘+’ and four ‘−’ signs can be arranged in a line such that no two ‘−’ signs occur together.
Prove that: 4nC2n : 2nCn = [1 · 3 · 5 ... (4n − 1)] : [1 · 3 · 5 ... (2n − 1)]2.
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
Write the expression nCr +1 + nCr − 1 + 2 × nCr in the simplest form.
Write the value of\[\sum^6_{r = 1} \ ^{56 - r}{}{C}_3 + \ ^ {50}{}{C}_4\]
Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.
Write the number of ways in which 5 red and 4 white balls can be drawn from a bag containing 10 red and 8 white balls.
Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.
