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In how many ways 5 rings can be worn on 3 fingers?
C) 60
Explanation: 0 0 0
Let these 3 circles are 3 fingers For 1st finger we have 5 choices, for second finger we have 4 choices left of rings, for third finger we have 3 choices left.
So total 5*4*3 = 60 ways
In how many ways the letters of the word ‘AUTHOR’ be arranged taking all the letters?
B) 720
Explanation: AUTHOR contains 6 letters, so total 6! ways.
In how many ways the letters of the word ‘MINIMUM’ be arranged taking all the letters?
A) 420
Explanation: MINIMUM contains 7 letters, so total 7! ways. But it contains 2 I’s and 3 M’s so divide by 2! And 3!
So ways 7!/(2! * 3!) = 7*6*5*4*3*2*1 / 2*1*3*2*1 = 420
How many words of 4 letters with or without meaning be made from the letters of the word ‘LEADING’, when repetition of letters is allowed?
D) 57624
Explanation: LEADING is 7 letters.
We have 4 places where letters are to be placed.
For first letter there are 7 choices, since repetition is allowed, for second, third and fourth letter also we have 7 choices each, so total of 7*7*7*7 ways = 2401 ways.
Now for arrangement of these 4 words, we have 4! Ways.
So total of 2401 * 4! Ways.
In how many ways letters of word ‘INVISIBLE’ be arranged such that all vowels are together?
B) 2880
Explanation:
First make IIIE in a circle. So we have Now we have N, V, S, B, L and box, their arrangements can be done in 6!
Letters inside circle are also to be arranged, we have I, I, I, E so ways are 4!/3!
Total ways 6! * 4!/3!
How many words can be made out of the letters of word ‘POUNDING’ such that all vowels occupy odd places?
A) 1440
Explanation: In POUNDING, there are 8 places 1 2 3 4 5 6 7 8
So for 3 places selection of vowels, we have 1, 3, 5, 7 number places ^{4}C_{3} ways Now for arranging these 3 vowels, ways are 3!
Remaining 5 are consonants (in which there are 2 N’s) for which 5!/2! so total ways = ^{4}C_{3} *3!*(5!/2!)
In how many ways a group of 2 men and 4 women be made out of a total of 4 men and 7 women?
B) 210
Explanation: We have to select 2 men from 4 men, and 4 women from 7 women So total ways = ^{4}C_{2} * ^{7}C_{4}
There are 8 men and 7 women. In how many ways a group of 5 people can be made such that at least 3 men are there in the group?
C) 1722
Explanation: Case 1: 3 men and 2 women ^{8}C_{3} * ^{7}C_{2} = 1176
Case 2: 4 men and 1 women ^{8}C_{4} * ^{7}C_{1} = 490
Case 3: all 5 men ^{8}C_{5 }= 56
Add all the cases.
There are 6 men and 7 women. In how many ways a committee of 4 members can be made such that a particular woman is always included.
D) 220
Explanation: There are total 13 people, a particular woman is to be included, so now 12 people are left to chosen from and 3 members to be chosen. So ways are ^{12}C_{3} .
There are 5 men and 3 women. In how many ways a committee of 3 members can be made such that 2 particular men are always to be excluded.
B) 20
Explanation: Total 8 people, 2 men are to excluded, so 6 men left to be chosen from and 3 members to be chosen. So ways are ^{6}C_{3} .
In a exam paper 1^{st} section contains 10 questions each with 5 choices and second section contains 5 questions each with 4 choices. In how manydifferent ways can the paper be answered if all the questions are attempted.
Answer – B. 5^{10} × 4^{5}
Explanation :
10 questions with 5 choices = 5^{10}
20 questions with 4 choices = 4^{5}
How many four digit number can be formed with the digits 5,9,1 and 3 only ?
Answer – C.256 Explanation : 4*4*4*4 = 256
Find the number of ways in which 9 students equally divided into three groups ?
Answer – A.280 Explanation : 9!/[3! ×(3!)^{3} ] = 362880/6*216 = 280
Find the number of ways of distributing 9 identical balls into 4 boxes so that no box is empty and each box being large enough to accommodate all balls ?
Answer – D.56 Explanation : ^{91}C_{41} = ^{8}C_{3} = 8*7*6/3*2*1 = 56
12 students participated in the competition and each get different score. In how many ways can three different prizes given ?
Answer – A.1320 Explanation : 12*11*10 = 1320
How many arrangement can be made from the word COMMERCE, such that all the vowels do not come together ?
Answer – D.9000 Explanation : 8 letters = 8!/ 2! 2! = 40320/4 = 10080 6 letters = 6!/2! = 360 Vowels = 3!/2! = 3 No of ways vowels together = 360*3 = 1080 No of ways vowels not together = 10080 – 1080 = 9000
Three brothers have 5 shirts, 8 pants and 6 ties. In how many ways can they wear them ?
Answer – C. 2419200 Explanation : ^{5}P_{3} * ^{8}P_{3} * ^{6}P_{3} = 60*336*120 = 2419200
5 men and 3 women are to be seated such that no 2 women sit together and 2 men sit together. Find the no of ways in which this can be arranged ?
Answer – B.720 Explanation : 5!*3! = 120*6 = 720
A group consists of 3 couples in which each of the 3 men have one wife each.In how many ways could they arranged in a straight line so that the men and women occupy alternate position ?
Answer – D.72 Explanation : 3!*3! + 3!*3! = 36+36 = 72
In how many ways can 5 different balls be distributed to 4 different boxes, when each of the can hold any no of ball ?
Answer – A.1024 Explanation : No of way =4^{5} = 1024
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