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Permutations and Combinations question

2016 · 9 Apr · Shift 1 · Q23
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Permutations and Combinations question

2016 · 9 Apr · Shift 1 · Q23

JEE MainMathematicsPermutations and CombinationsMCQ+4 / −1
If the four letter words (need not be meaningful ) are to be formed using the letters from the word “MEDITERRANEAN” such that the first letter is R and the fourth letter is E, then the total number of all such words is :
  1. A
    11!(2!)3{{11!} \over {{{\left( {2!} \right)}^3}}}(2!)311!​
  2. B
    110
  3. C
    56
  4. D
    59
View written solutionFree

Correct answer: D

  1. Count the letters in MEDITERRANEAN

    The word is: M E D I T E R R A N E A N\text{M E D I T E R R A N E A N}M E D I T E R R A N E A N

    Frequency of letters:

    • E=3E = 3E=3
    • R=2R = 2R=2
    • A=2A = 2A=2
    • N=2N = 2N=2
    • M,D,I,T=1M,D,I,T = 1M,D,I,T=1 each
  2. Apply the position restrictions

    We need 4-letter words such that:

    • first letter is RRR
    • fourth letter is EEE

    So the form is: R__ER\_\_ER__E

    After fixing one RRR and one EEE, the remaining available letters are:

    • E=2E = 2E=2
    • R=1R = 1R=1
    • A=2A = 2A=2
    • N=2N = 2N=2
    • M,D,I,T=1M,D,I,T = 1M,D,I,T=1 each

    Thus, for the 2nd and 3rd positions, we must choose an ordered pair from this remaining multiset.

  3. Count all possible ordered pairs for positions 2 and 3

    Distinct available letter types are: {E,R,A,N,M,D,I,T}\{E,R,A,N,M,D,I,T\}{E,R,A,N,M,D,I,T} i.e. 888 types.

    We count ordered pairs in two cases.

    Case 1: Both letters different

    • Choose the 2nd letter in 888 ways.
    • Choose the 3rd letter in 777 ways.

    Total: 8×7=568 \times 7 = 568×7=56

    Case 2: Both letters same

    This is possible only for letters having at least 2 copies left.

    Such letters are: E,A,NE, A, NE,A,N

    (RRR has only 1 left, so RRRRRR is not possible.)

    Hence same-letter pairs possible are: EE,AA,NNEE, AA, NNEE,AA,NN

    Total: 333

  4. Total number of words

    56+3=5956 + 3 = 5956+3=59

  5. Check options

    • A: not equal to 595959
    • B: 110110110 ❌
    • C: 565656 ❌
    • D: 595959 ✅

Therefore, the correct answer is: 59\boxed{59}59​

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