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Quiz: Permutations and Combinations

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Number of Questions: 40

Question: 6 -

In a colony, there are 55 members. Every member posts a greeting card to all the members. How many greeting cards were posted by them?

Options:

  1. 890

  2. 990

  3. 2970

  4. 1980

    5. Answer:

    2970
    Solution:

    First player can post greeting cards to the remaining 54 players in 54 ways. Second player can post greeting card to the 54 players. Similarly, it happens with the rest of the players. The total numbers of greeting cards posted are
    54 + 54 + 54 …
    54 (55times) = 54 x 55 = 2970. 

Question: 7 -

If nPr = 3024 and nCr = 126 then find n and r.

Options:

  1. 11, 4

  2. 9, 4

  3. 10, 3

  4. 12, 4

    5. Answer:

    9, 4
    Solution:

    nPrnCr=3024126
    nPr=n!(n−r)!
    nCr=n!(n−r)!×r!
    Hence [n!(n−r)!]÷[n!(n−r)!×r!] = 24
    24 = r!
    Hence r = 4
    Now nP4 = 3024
    n!(n−4)!=3024
    n(n-1)(n-2)(n-3) = 9.8.7.6
    n = 9.

Question: 8 -

A committee has 5 men and 6 women. What are the number of ways of selecting 2 men and 3 women from the given committee? 

Options:

  1. 150

  2. 300

  3. 250

  4. 200

    5. Answer:

    200
    Solution:

    The number of ways to select two men and three women = 
    = (5 *4 )/(2 * 1) * (6 * 5 * 4)/(3 * 2)
    = 200

Question: 9 -

Find the number of ways of arranging the letters of the words DANGER, so that no vowel occupies odd place.

Options:

  1. 96

  2. 144

  3. 48

  4. 36

    5. Answer:

    144
    Solution:

    The given word is DANGER. Number of letters is 6. Number of vowels is 2 (i.e., A, E). Number of consonants is 4 (i.e., D,N,G,R). As the vowels cannot occupy odd places, they can be arranged in even places. Two vowels can be arranged in 3 even places in 3P2 ways i.e., 6. Rest of the consonants can arrange in the remaining 4 places in 4! ways. The total number of arrangements is 6 x 4! = 144.

Question: 10 -

The number of permutations of the letters of the word 'MESMERISE' is_. 

Options:

  1. 9!/(2!)2 (3!)2

  2. 9!/(2!)3 3!

  3. 9!/(2!)2 3!

  4. 5!/(2!)2 3!

    5. Answer:

    9!/(2!)2 3!
    Solution:

    n items of which p are alike of one kind, q alike of the other, r alike of another kind and the remaining are distinct can be arranged in a row in n!/p!q!r! ways.

    The letter pattern 'MESMERISE' consists of 10 letters of which there are 2M's, 3E's, 2S's and 1I and 1R.
    Number of arrangements = 9!/(2!)2 3!