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In how many ways can the letters of the word DELHI be arranged?
A and B can occupy the first and the ninth places, the second and the tenth places, the third and the eleventh place and so on... This can be done in 18 ways.
A and B can be arranged in 2 ways.
All the other 24 alphabets can be arranged in 24! ways.
Hence the required answer = 2 x 18 x 24!
How many numbers between 200 and 1200 can be formed with the digits 0, 1, 2, 3 (repetition of digits not allowed?
14 numbers
So total of 14 numbers (without repetition) between 200 & 1200 with 0, 1, 2 & 3
^{28}C_{2r}/^{24}C_{2r4} = 225/11
⇒ 28!/2r!(282r)! * [(2r4)!(282r)!]/24! = 225/11
⇒ (28*27*26*25)/[2r*(2r1)*(2r2)*(2r3)] = 225/11
⇒ 2r*(2r1)*(2r2)*(2r3) = (28*27*26*25*11)/225
⇒ 2r*(2r1)*(2r2)*(2r3) = 28*3*26*11
⇒ 2r*(2r1)*(2r2)*(2r3) = 4*7*3*13*2*11
⇒ 2r*(2r1)*(2r2)*(2r3) = (2*7)*13*(3*4)*11
⇒ 2r*(2r1)*(2r2)*(2r3) = 14*13*12*11
⇒ 2r = 14
⇒ r = 7
In how many ways can 10 identical presents be distributed among 6 children so that each child gets at least one present?
We have to count natural numbers which have a maximum of 4 digits. The required answer will be given by: Number of single digit numbers + Number of two digit numbers + Number of three digit numbers + Number of four digit numbers.
A captain and a vicecaptain are to be chosen out of a team having eleven players. How many ways are there to achieve this?
Explaination: out of 11 player 1 captain can be choose 11 ways, Now remaining 10 player,wise captain can be choose in 10 ways Therefore total number of ways =11*10=110 ways
In how many ways can Ram choose a vowel and a consonant from the letters of the word ALLAHABAD?
In the ALLAHABAD :
only 1 vowel available for selection (A).
A is available 4 times .
there are 4 consonants available – L, H, B, D
Then the number of ways of selecting a vowel and a consonant would be 1 × ^{4}C_{1} = 4.
How many motor vehicle registration number of 4 digits can be formed with the digits 0, 1,2, 3, 4, 5? (No digit being repeated.)
The correct option is Option D.
There are six different digits (0,1,2,3,4,5) and these will be arranged in 6p4 ways = 360
Hence, 360 different registration numbers can be formed with the help of these numbers.
There are ten subjects in the school day at St.Vincent’s High School but the sixth standard students have only 5 periods in a day. In how many ways can we form a time table for the day for the sixth standard students if no subject is repeated?
There are 10 subjects and 5 periods
I st period can be filled with any 10 subjects
2nd period can be filled with remaining 9 subjects (1 subject is already filled)
3 rd period can be filled with remaining 8 subjects (2 subjects are already filled)
4 th period can be filled with remaining 7 subjects (3 subjects are already filled)
5 rd period can be filled with remaining 6 subjects (4 subjects are already filled)
so total no fo ways we can arrange = 10*9*8*7*6
= 30240 ways
How many batting orders are possible for the Indian cricket team if there is a squad of 15 to choose from such that Sachin Tendulkar is always chosen?
The selection of the II player team can be done in ^{14}C_{10} ways. This results in the team of 11 players being completely chosen. The arrangements of these 11 players can be done in 11!.
Total batting orders = ^{14}C_{10} x 11! = 1001 x 11!
(Note: Arrangement is required here because we are talking about forming batting orders).
How many distinct words can be formed out of the word PROWLING which start with R & end with W?
How many even numbers of four digits can be formed with the digits 1, 2, 3, 4, 5, 6 (repetitions of digits are allowed)?
for an even number, the ones place number should be even i.e it could be either one of 2,4,6
so number of ways to select ones place number is ^{3}C_{1} = 3
since repetetion is allowed, and all numbers are whole numbers.
so number of ways to select any of the three remaining numbers = ^{6}C_{1} =6
Number of 4 digit number possible = 6 x 6 x 6 x 3
= 648
On a shelf, 2 books of Geology, 2 books of Sociology and 5 of Economics are to be arranged in such a way that the books of any subject are to be together. Find in how many ways can this be done?
There are books of 3 subjects (Geology, Sociology and Economics), hence they can be arranged in 3! (3 * 2 * 1) = 6 ways.
Further, in each category (subject), books are to be arranged in different order, we get,
Required number of ways:
3! * [2! * 2! * 5!] = 2880
In how many ways can the letters of the word ‘EQUATION’ be arranged so that all the vowels come together?
The correct option is Option B.
The word equation has 8 letters of which 5 are vowels and 3 are consonants. Bunch up the 5 vowels to assume them as one letter. Thus the total number of letters is 4 which can be arranged in 4!=24 . And for each of these arrangements the 5 vowels can be arranged among themselves in 5!=120 ways. Thus, the total arrangements of the letters with the vowels always appearing together is 24∗120=2880 .
How many quadrilateral can be formed from 25 points out of which 7 are collinear
otal number of quadrilateral combination possible if none of the points are collinear = ^{25}C_{4} = 12650
If we form a geometry by joining any three points out of seven collinear points and one point from 18 non collinear points, it will give us a triangle instead of quadrilateral. So we have to eliminate number of combinations which can be formed in this way, which is ^{7}C_{3} x ^{18}C_{1} = 35 x 18 = 630
We also can't form quadrilateral if we choose all four vertices of quadrilateral to be any 4 points from 7 collinear points. It will come out to be a straight line. So we have to eliminate such combinations also. Which is ^{7}C_{4} = 35
So net number of possible quadrilaterals = 12650  630  35 = 11985
In how many ways a committee consisting of 5 men and 3 women, can be chosen from 9 men and 12 women.
Choose 5 men out of 9 men = ^{9}C_{5} ways = 126 ways
Choose 3 women out of 12 women = ^{12}C_{3} ways = 220 ways
The committee can be chosen in 27720 ways
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