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
Convert the following products into factorials:
3 · 6 · 9 · 12 · 15 · 18
Prove that: n! (n + 2) = n! + (n + 1)!
If (n + 1)! = 90 [(n − 1)!], find n.
If (n + 3)! = 56 [(n + 1)!], find n.
Prove that:
If P (n, 5) = 20. P(n, 3), find n ?
If P (15, r − 1) : P (16, r − 2) = 3 : 4, find r.
If n +5Pn +1 =\[\frac{11 (n - 1)}{2}\]n +3Pn, find n.
Find the number of different 4-letter words, with or without meanings, that can be formed from the letters of the word 'NUMBER'.
There are two works each of 3 volumes and two works each of 2 volumes; In how many ways can the 10 books be placed on a shelf so that the volumes of the same work are not separated?
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?
How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
How many words can be formed from the letters of the word 'SUNDAY'? How many of these begin with D?
How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:
the letters P and I respectively occupy first and last place?
How many permutations can be formed by the letters of the word, 'VOWELS', when
each word begins with E?
How many permutations can be formed by the letters of the word, 'VOWELS', when
all consonants come together?
In how many ways can a lawn tennis mixed double be made up from seven married couples if no husband and wife play in the same set?
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.
Find the number of words formed by permuting all the letters of the following words:
PAKISTAN
Find the total number of arrangements of the letters in the expression a3 b2 c4 when written at full length.
How many words can be formed with the letters of the word 'PARALLEL' so that all L's do not come together?
How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?
In how many ways can the letters of the word 'ARRANGE' be arranged so that the two R's are never 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.
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?
If the permutations of a, b, c, d, e taken all together be written down in alphabetical order as in dictionary and numbered, find the rank of the permutation debac ?
In how many ways can the letters of the word "INTERMEDIATE" be arranged so that:
the relative order of vowels and consonants do not alter?
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 expression nCr +1 + nCr − 1 + 2 × nCr in the simplest form.
Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.
Write the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants.
