More practice questions

  1. Q5. Determine n if (i) 2nC3 : nC3 = 12 : 1; (ii) 2nC3 : nC3 = 11 : 1 (NCERT Exercise 6.4, Q2).

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    The ratio is 2n(2n − 1)(2n − 2)/(n(n − 1)(n − 2)) = 4(2n − 1)/(n − 2) (the 3! cancels, and 2n − 2 = 2(n − 1)). (i) 4(2n − 1) = 12(n − 2) gives n = 5. (ii) 4(2n − 1) = 11(n − 2) gives n = 6.

  2. Q6. How many ways are there to select 9 balls from 6 red, 5 white and 5 blue balls if each selection has 3 balls of each colour? Treat all balls as individually distinguishable, including balls of the same colour (NCERT Exercise 6.4, Q5).

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    6C3 × 5C3 × 5C3 = 20 × 10 × 10 = 2000.

  3. Q7. How many 5-card hands from a pack of 52 contain exactly one ace (NCERT Exercise 6.4, Q6)?

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    Choose the ace in 4 ways and the other 4 cards from the 48 non-aces: 4 × 48C4 = 4 × 194 580 = 778 320.

  4. Q8. A cricket team of 11 is chosen from 17 players, of whom only 5 can bowl. In how many ways if the team must include exactly 4 bowlers (NCERT Exercise 6.4, Q7)?

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    5C4 × 12C7 = 5 × 792 = 3960.

  5. Q9. A committee of 7 is formed from 9 boys and 4 girls. In how many ways if it has (i) exactly 3 girls; (ii) at least 3 girls; (iii) at most 3 girls (NCERT Miscellaneous Exercise, Q3)?

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    (i) 4C3 × 9C4 = 4 × 126 = 504. (ii) 504 + 4C4 × 9C3 = 504 + 84 = 588. (iii) 0, 1, 2 or 3 girls: 9C7 + 4 × 9C6 + 6 × 9C5 + 504 = 36 + 336 + 756 + 504 = 1632 (or 13C7 − 84 = 1716 − 84).

  6. Q10. From a class of 25, 10 go on an excursion; 3 particular students will go only all together or not at all. In how many ways can the party be chosen (NCERT Miscellaneous Exercise, Q10)?

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    All three go: choose 7 more from 22, 22C7 = 170 544. None go: 22C10 = 646 646. Total 817 190.

  7. Q11. A bag contains 5 black and 6 red balls. In how many ways can 2 black and 3 red balls be selected? Treat all balls as individually distinguishable, including balls of the same colour (NCERT Exercise 6.4, Q8).

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    5C2 × 6C3 = 10 × 20 = 200.