Mathematics Mcqs

In how many different ways can six players be arranged in a line such that two of them, Asim and Raheem are never together?

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In how many different ways can six players be arranged in a line such that two of them, Asim and Raheem are never together?

A. 120
B. 240
C. 360
D. 480
Explanation:
1. As there are six players, So total ways in which they can be arranged = 6!ways =720.
A number of ways in which Asim and Raheem are together = 5!x2 = 240.
Therefore, Number of ways when they don’t remain together = 720 -240 =480.

Groups each containing 3 boys are to be formed out of 5 boys. A, B, C, D and E such that no group can contain both C and D together. What is the maximum number of such different groups?

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Groups each containing 3 boys are to be formed out of 5 boys. A, B, C, D and E such that no group can contain both C and D together. What is the maximum number of such different groups?

A. 5
B. 6
C. 7
D. 8
Explanation:
Maximum number of such different groups = ABC, ABD,ABE, BCE,BDE,CEA,DEA =7.
Alternate method:
Total number of way in which 3 boys can be selected out of 5 is 5C3
Number of ways in which CD comes together = 3 (CDA,CDB,CDE)
Therefore, Required number of ways = 5C3 -3
= 10-3 =7.

A question paper had 10 questions. Each question could only be answered as true(T) of False(F). Each candidate answered all the questions, Yet no two candidates wrote the answers in an identical sequence. How many different sequences of answers are possible?

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A question paper had 10 questions. Each question could only be answered as true(T) of False(F). Each candidate answered all the questions, Yet no two candidates wrote the answers in an identical sequence. How many different sequences of answers are possible?

A. 20
B. 40
C. 512
D. 1024
Each question can be answered in 2 ways.
10 Questions can be answered = 210= 1024 ways.

Six points are marked on a straight line and five points are marked on another line which is parallel to the first line. How many straight lines, including the first two, can be formed with these points?

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Six points are marked on a straight line and five points are marked on another line which is parallel to the first line. How many straight lines, including the first two, can be formed with these points?

A. 29
B. 32
C. 55
D. 30
We know that, the number of straight lines that can be formed by the 11 points in which 6 points are collinear and no other set of three points, except those that can be selected out of these 6 points are collinear.
Hence, the required number of straight lines
= ¹¹C₂ – ⁶C₂ – ⁵C₂ + 1 + 1
= 55 – 15 – 10 + 2 = 32

A group of 10 representatives is to be selected out of 12 seniors and 10 juniors. In how many different ways can the group be selected if it should have at least one senior?

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A group of 10 representatives is to be selected out of 12 seniors and 10 juniors. In how many different ways can the group be selected if it should have at least one senior?

A. ²²C₁₀
B. ²²C₁₀ + 1
C. ²²C₉ + ¹⁰C₁
D. ²²C₁₀ – 1
Explanation:
The total number of ways of forming the group of ten representatives is ²²C₁₀.
The total number of ways of forming the group that consists of no seniors is ¹⁰C₁₀ = 1 way
The required number of ways = ²²C₁₀ – 1

A group of 10 representatives is to be selected out of 12 seniors and 10 juniors. In how many different ways can the group be selected, if it should have 5 seniors and 5 juniors?

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A group of 10 representatives is to be selected out of 12 seniors and 10 juniors. In how many different ways can the group be selected, if it should have 5 seniors and 5 juniors?

A. ¹²C₅ * 10
B. ¹²C₇ * 10
C. ¹²C₇ * ¹⁰C₅
D. 12 * ¹⁰C₅
Here, five seniors out of 12 seniors can be selected in ¹²C₅ ways. Also, five juniors out of ten juniors can be selected ¹⁰C₅ ways. Hence the total number of different ways of selection = ¹²C₅ * ¹⁰C₅ = ¹²C₇ * ¹⁰C₅
= ¹²C₅ = ¹²C₇

A group consists of 4 men, 6 women and 5 children. In how many ways can 3 men, 2 women and 3 children selected from the given group?

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A group consists of 4 men, 6 women and 5 children. In how many ways can 3 men, 2 women and 3 children selected from the given group?

A. 300
B. 450
C. 600
D. 750
Explanation:
The number of ways of selecting three men, two women and three children is:
= ⁴C₃ * ⁶C₂ * ⁵C₃
= (4 * 3 * 2)/(3 * 2 * 1) * (6 * 5)/(2 * 1) * (5 * 4 * 3)/(3 * 2 * 1)
= 4 * 15 * 10
= 600 ways.

A group consists of 4 men, 6 women and 5 children. In how many ways can 2 men , 3 women and 1 child selected from the given group?

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A group consists of 4 men, 6 women and 5 children. In how many ways can 2 men , 3 women and 1 child selected from the given group?

A. 300
B. 600
C. 750
D. 900
Explanation:
Two men, three women and one child can be selected in ⁴C₂ * ⁶C₃ * ⁵C₁ ways
= (4 * 3)/(2 * 1) * (6 * 5 * 4)/(3 * 2) * 5
= 600 ways.

Find the number of ways of arranging the letters of the word “MATERIAL” such that all the vowels in the word are to come together?

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Find the number of ways of arranging the letters of the word “MATERIAL” such that all the vowels in the word are to come together?

A. 720
B. 1440
C. 1860
D. 2160
In the word, “MATERIAL” there are three vowels A, I, E.
If all the vowels are together, the arrangement is MTRL’AAEI’.
Consider AAEI as one unit. The arrangement is as follows.
M T R L A A E I
The above 5 items can be arranged in 5! ways and AAEI can be arranged among themselves in 4!/2! ways.
Number of required ways of arranging the above letters = 5! * 4!/2!
= (120 * 24)/2 = 1440 ways.