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Permutation and Combination Questions Answers

  • 8. In how many words can be formed by using all letters of the word BHOPAL

    1. 420
    2. 520
    3. 620
    4. 720
    Answer And Explanation

    Answer: Option D

    Explanation:

    Required number
    \begin{aligned}
    = 6! \\
    = 6*5*4*3*2*1 \\
    = 720
    \end{aligned}

  • 9. In how many way the letter of the word "APPLE" can be arranged

    1. 20
    2. 40
    3. 60
    4. 80
    Answer And Explanation

    Answer: Option C

    Explanation:

    Friends the main point to note in this question is letter "P" is written twice in the word.
    Easy way to solve this type of permutation question is as,

    So word APPLE contains 1A, 2P, 1L and 1E
    Required number =
    \begin{aligned}
    = \frac{5!}{1!*2!*1!*1!} \\
    = \frac{5*4*3*2!}{2!} \\
    = 60
    \end{aligned}

  • 10. In how many ways can the letters of the CHEATER be arranged

    1. 20160
    2. 2520
    3. 360
    4. 80
    Answer And Explanation

    Answer: Option B

    Explanation:

    As we can see the letter "E" is twice in given word, so Required Number
    \begin{aligned}
    = \frac{7!}{2!} \\
    = \frac{7*6*5*4*3*2!}{2!} \\
    = 2520
    \end{aligned}

  • 11. In how many way the letter of the word "RUMOUR" can be arranged

    1. 2520
    2. 480
    3. 360
    4. 180
    Answer And Explanation

    Answer: Option D

    Explanation:

    In above word, there are 2 "R" and 2 "U"
    So Required number will be

    \begin{aligned}
    = \frac{6!}{2!*2!} \\
    = \frac{6*5*4*3*2*1}{4} \\
    = 180
    \end{aligned}

  • 12. How many words can be formed from the letters of the word "SIGNATURE" so that vowels always come together.

    1. 17280
    2. 4320
    3. 720
    4. 80
    Answer And Explanation

    Answer: Option A

    Explanation:

    word SIGNATURE contains total 9 letters.
    There are four vowels in this word, I, A, U and E

    Make it as, SGNTR(IAUE), consider all vowels as 1 letter for now
    So total letter are 6.
    6 letters can be arranged in 6! ways = 720 ways
    Vowels can be arranged in themselves in 4! ways = 24 ways

    Required number of ways = 720*24 = 17280

  • 13. In how many ways can the letters of the word "CORPORATION" be arranged so that vowels always come together.

    1. 5760
    2. 50400
    3. 2880
    4. None of above
    Answer And Explanation

    Answer: Option B

    Explanation:

    Vowels in the word "CORPORATION" are O,O,A,I,O
    Lets make it as CRPRTN(OOAIO)

    This has 7 lettes, where R is twice so value = 7!/2!
    = 2520

    Vowel O is 3 times, so vowels can be arranged = 5!/3!

    = 20

    Total number of words = 2520 * 20 = 50400

  • 14. In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there

    1. 109
    2. 128
    3. 138
    4. 209
    Answer And Explanation

    Answer: Option D

    Explanation:

    In a group of 6 boys and 4 girls, four children are to be selected such that
    at least one boy should be there.
    So we can have

    (four boys) or (three boys and one girl) or (two boys and two girls) or (one boy and three gils)

    This combination question can be solved as

    \begin{aligned}
    (^{6}{C}_{4}) + (^{6}{C}_{3} * ^{4}{C}_{1}) + \\
    + (^{6}{C}_{2} * ^{4}{C}_{2}) + (^{6}{C}_{1} * ^{4}{C}_{3}) \\

    = \left[\dfrac{6 \times 5 }{2 \times 1}\right] + \left[\left(\dfrac{6 \times 5 \times 4 }{3 \times 2 \times 1}\right) \times 4\right] + \\\left[\left(\dfrac{6 \times 5 }{2 \times 1}\right)\left(\dfrac{4 \times 3 }{2 \times 1}\right)\right] + \left[6 \times 4 \right] \\
    = 15 + 80 + 90 + 24\\
    = 209
    \end{aligned}