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Permutations and Combinations question

2021 · 24 Feb · Shift 1 · Q29
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  5. /2021 · 24 Feb · Shift 1 · Q29

Permutations and Combinations question

2021 · 24 Feb · Shift 1 · Q29

JEE MainMathematicsPermutations and CombinationsMCQ+4 / −1
A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :
  1. A
    1050
  2. B
    575
  3. C
    560
  4. D
    1625
View written solutionFree

Correct answer: D

  1. Let the number of Indians selected be iii and the number of foreigners selected be fff.

  2. The conditions are:

    • At least 222 Indians: i≥2i \ge 2i≥2
    • Number of foreigners is double the number of Indians: f=2if = 2if=2i

  3. Since there are only 666 Indians and 888 foreigners available:

    • i≤6i \le 6i≤6
    • f=2i≤8⇒i≤4f = 2i \le 8 \Rightarrow i \le 4f=2i≤8⇒i≤4

    So possible values of iii are: i=2,3,4i = 2, 3, 4i=2,3,4

  4. Count the number of committees for each case.

    Case 1: i=2i=2i=2, then f=4f=4f=4 (62)(84)=15×70=1050\binom{6}{2}\binom{8}{4} = 15 \times 70 = 1050(26​)(48​)=15×70=1050

    Case 2: i=3i=3i=3, then f=6f=6f=6 (63)(86)=20×28=560\binom{6}{3}\binom{8}{6} = 20 \times 28 = 560(36​)(68​)=20×28=560

    Case 3: i=4i=4i=4, then f=8f=8f=8 (64)(88)=15×1=15\binom{6}{4}\binom{8}{8} = 15 \times 1 = 15(46​)(88​)=15×1=15

  5. Total number of ways: 1050+560+15=16251050 + 560 + 15 = 16251050+560+15=1625

  6. Therefore, the required number of committees is: 1625\boxed{1625}1625​

  7. Comparing with the stored correct answer:

    • Stored correct answer: DDD
    • Our derived answer: 162516251625, which is option DDD

Hence, the stored answer is correct.

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