More practice questions

  1. Q5. If 1/6! + 1/7! = x/8!, find x. Evaluate n!/(n − r)! for (i) n = 6, r = 2; (ii) n = 9, r = 5 (NCERT Exercise 6.2, Q4 and Q5).

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    8! = 56 × 6! = 8 × 7!, so 1/6! = 56/8! and 1/7! = 8/8!: x = 64. (i) 6 × 5 = 30; (ii) 9 × 8 × 7 × 6 × 5 = 15 120.

  2. Q6. How many 3-digit even numbers can be made from 1, 2, 3, 4, 6, 7 with no digit repeated (NCERT Exercise 6.3, Q3)?

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    The unit’s place is 2, 4 or 6 (3 ways); the other two places then use 2 of the remaining 5 digits in order: 5 × 4 = 20. So 3 × 20 = 60.

  3. Q7. Find n if n−1P3 : nP4 = 1 : 9, and r if (i) 5Pr = 2 × 6Pr−1; (ii) 5Pr = 6Pr−1 (NCERT Exercise 6.3, Q6 and Q7).

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    nP4 = n × n−1P3, so the ratio is 1/n = 1/9 and n = 9. For r: 5!/(5 − r)! = 2 × 6!/(7 − r)! gives (7 − r)(6 − r) = 12, so r = 3 or r = 10; only r = 3 is allowed (r ≤ 5). (ii) 5!/(5 − r)! = 6!/(7 − r)! gives (7 − r)(6 − r) = 6, so r = 4 or r = 9; only r = 4 is allowed.

  4. Q8. Using MONDAY with no letter repeated, how many words (i) use 4 letters; (ii) use all 6; (iii) use all 6 and start with a vowel (NCERT Exercise 6.3, Q9)?

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    (i) 6P4 = 360; (ii) 6! = 720; (iii) the first letter is O or A (2 ways), then 5! = 120 for the rest: 240.

  5. Q9. In how many ways can the letters of PERMUTATIONS be arranged if (i) the words start with P and end with S; (ii) the vowels are all together; (iii) there are always 4 letters between P and S (NCERT Exercise 6.3, Q11)?

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    12 letters with T twice. (i) 10!/2! = 1 814 400. (ii) The 5 vowels E, U, A, I, O as one block with 7 other letters: 8!/2! × 5! = 2 419 200. (iii) P and S can be in 7 position pairs, in 2 orders, and the other 10 letters in 10!/2! ways: 14 × 1 814 400 = 25 401 600.

  6. Q10. How many 3-digit numbers can be formed from the digits 1 to 9 with no digit repeated? How many 4-digit numbers can be formed from 1, 2, 3, 4, 5 with no digit repeated, and how many of them are even? In how many ways can a chairman and a vice chairman be chosen from a committee of 8 (NCERT Exercise 6.3, Q1, Q4 and Q5)?

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    9P3 = 9 × 8 × 7 = 504. 5P4 = 120; an even number ends in 2 or 4 (2 ways), and the other three places take 3 of the remaining 4 digits in order (24 ways), so 2 × 24 = 48 are even. 8P2 = 8 × 7 = 56.

  7. Q11. How many 3-letter words from the letters of NUMBER have at least one letter repeated?

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    All words with repetition allowed, minus those with no letter repeated: 63 − 6P3 = 216 − 120 = 96.