Advertisements
Advertisements
Question
Evaluate
Advertisements
Solution
Given:
\[{}^{20} C_5 + \sum^5_{r = 2} 25 -^r C_4\]
\[ = {}^{20} C_5 +^{23} C_4 + {}^{22} C_4 + {}^{21} C_4 + {}^{20} C_4 \]
\[ = \left( {}^{20} C_{4_{}} + {}^{20} C_5 \right) +^{21} C_4 +^{22} C_4 +^{23} C_4 \]
\[=^{21} C_5 +^{21} C_4 + {}^{{}^{22}} C_4 +^{23} C_4\]
[∵\[{{}^{}}^n C_{r - 1} +^n C_r =^{n + 1} C_r\]]
\[= \left( {}^{21} C_{4_{}} + {}^{21} C_{5_{}} \right) +^{22} C_4 +^{23} C_4 \]
\[ =^{24} C_5\] [∵ \[{{}^{}}^n C_{r - 1} +^n C_r =^{n + 1} C_r\]]
APPEARS IN
RELATED QUESTIONS
Prove that: n! (n + 2) = n! + (n + 1)!
If (n + 1)! = 90 [(n − 1)!], find n.
If \[\frac{(2n)!}{3! (2n - 3)!}\] and \[\frac{n!}{2! (n - 2)!}\] are in the ratio 44 : 3, find n.
If nP4 = 360, find the value of n.
If P (2n − 1, n) : P (2n + 1, n − 1) = 22 : 7 find n.
If P (n, 5) : P (n, 3) = 2 : 1, find n.
If P (15, r − 1) : P (16, r − 2) = 3 : 4, 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?
How many words, with or without meaning, can be formed by using the letters of the word 'TRIANGLE'?
How many three-digit numbers are there, with no digit repeated?
In how many ways can 6 boys and 5 girls be arranged for a group photograph if the girls are to sit on chairs in a row and the boys are to stand in a row behind them?
In how many ways can the letters of the word 'STRANGE' be arranged so that
the vowels occupy only the odd places?
How many different words can be formed with the letters of word 'SUNDAY'? How many of the words begin with N? How many begin with N and end in Y?
How many permutations can be formed by the letters of the word, 'VOWELS', when
all vowels come together?
How many permutations can be formed by the letters of the word, 'VOWELS', when
all consonants come together?
Find the number of words formed by permuting all the letters of the following words:
INDEPENDENCE
Find the number of words formed by permuting all the letters of the following words:
CONSTANTINOPLE
In how many ways can the letters of the word 'ALGEBRA' be arranged without changing the relative order of the vowels and consonants?
How many words can be formed with the letters of the word 'PARALLEL' so that all L's do not come together?
How many permutations of the letters of the word 'MADHUBANI' do not begin with M but end with I?
Find the number of numbers, greater than a million, that can be formed with the digits 2, 3, 0, 3, 4, 2, 3.
How many different arrangements can be made by using all the letters in the word 'MATHEMATICS'. How many of them begin with C? How many of them begin with T?
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?
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.
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 the product of 2n consecutive negative integers is divisible by (2n)!
Prove that: 4nC2n : 2nCn = [1 · 3 · 5 ... (4n − 1)] : [1 · 3 · 5 ... (2n − 1)]2.
Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
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 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 number of diagonals of an n-sided polygon.
Write the maximum number of points of intersection of 8 straight lines in a plane.
