Tips to solve Permutation and Combination Problems

  1. Factorial Notation:Let n be a positive integer. Then, factorial n, denoted n! is defined as:

    n! = n(n – 1)(n – 2) … 3.2.1.

    Examples:

    1. We define 0! = 1.
    2. 4! = (4 x 3 x 2 x 1) = 24.
    3. 5! = (5 x 4 x 3 x 2 x 1) = 120.
  2. Permutations:The different arrangements of a given number of things by taking some or all at a time, are called permutations.Examples:
    1. All permutations (or arrangements) made with the letters abc by taking two at a time are (abbaaccabccb).
    2. All permutations made with the letters abc taking all at a time are:
      ( abcacbbacbcacabcba)
  3. Number of Permutations:Number of all permutations of n things, taken r at a time, is given by:
    nPr = n(n – 1)(n – 2) … (n – r + 1) = n!
    (n – r)!

    Examples:

    1. 6P2 = (6 x 5) = 30.
    2. 7P3 = (7 x 6 x 5) = 210.
    3. Cor. number of all permutations of n things, taken all at a time = n!.
  4. An Important Result:If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind; p3 are alike of third kind and so on and pr are alike of rth kind,
    such that (p1 + p2 + … pr) = n.

    Then, number of permutations of these n objects is = n!
    (p1!).(p2)!…..(pr!)
  5. Combinations:Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.Examples:
    1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.Note: AB and BA represent the same selection.
    2. All the combinations formed by abc taking abbcca.
    3. The only combination that can be formed of three letters abc taken all at a time is abc.
    4. Various groups of 2 out of four persons A, B, C, D are:

      AB, AC, AD, BC, BD, CD.

    5. Note that ab ba are two different permutations but they represent the same combination.
  6. Number of Combinations:The number of all combinations of n things, taken r at a time is:
    nCr = n! = n(n – 1)(n – 2) … to r factors .
    (r!)(n – r)! r!

    Note:

    1. nCn = 1 and nC0 = 1.
    2. nCr = nC(n – r)

    Examples:

    i.   11C4 = (11 x 10 x 9 x 8) = 330.
    (4 x 3 x 2 x 1)
    ii.   16C13 = 16C(16 – 13) = 16C3 = 16 x 15 x 14 = 16 x 15 x 14 = 560.
    3! 3 x 2 x 1
    From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?
    A. 564
    B. 645
    C. 735
    D. 756
    E. None of these

    Answer: Option D

    Explanation:

    We may have (3 men and 2 women) or (4 men and 1 woman) or (5 men only).

     Required number of ways = (7C3 x 6C2) + (7C4 x 6C1) + (7C5)
    = 7 x 6 x 5 x 6 x 5 + (7C3 x 6C1) + (7C2)
    3 x 2 x 1 2 x 1
    = 525 + 7 x 6 x 5 x 6 + 7 x 6
    3 x 2 x 1 2 x 1
    = (525 + 210 + 21)
    In how many different ways can the letters of the word ‘LEADING’ be arranged in such a way that the vowels always come together?
    A. 360
    B. 480
    C. 720
    D. 5040
    E. None of these

    Answer: Option C

    Explanation:

    The word ‘LEADING’ has 7 different letters.

    When the vowels EAI are always together, they can be supposed to form one letter.

    Then, we have to arrange the letters LNDG (EAI).

    Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways.

    The vowels (EAI) can be arranged among themselves in 3! = 6 ways.

     Required number of ways = (120 x 6) = 720.

    = 756.

     

    In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?

    Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

    In how many ways can the letters of the word 'LEADER' be arranged?

    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?

    How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

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