Combinatorics practice problems

Q. Num words from ‘LAUGH’ letters if vowels together? => Treat AU as one letter and finally multiply by 2! => 4*2! = 48


Find number of ways of arrange 5 Boys, 5 Girls

  1. No restrictions? 10!

  2. All G together 6! * 5!

  3. All G together, all B together 2! * 5! * 5!

  4. No 2 G together 5! (arrange boys) * 6C5(Choose 5 slots from 6) * 5!(permute girls)

  5. No 2 G together, No 2 B together 5! * 5! * 2(Only 2 ways to choose slots now)

  6. B1B2 together, G1G2 together 8! * 2! * 2!

  7. B1B2 separate, G1G2 together First ignore B1, B2 => 7! * 2! Now there are 8 slots to choose from: 8C2 Permute boys: 2! Finally: 7! * 2! * 8C2 * 2!

  8. B1B2 separate, G1G2 separate Total - (G1G2 together, B1B2 together - G1G2 together B1B2 separate - vice versa) 10! - 8! 2! 2! - 7! * 2! * 8C2 * 2! - 7! * 2! * 8C2 * 2! = 8! ( 90 - 4 - 7*2 - 7*2) = 8! (58)


How many words with letters of ‘DAUGHTER’ if
1. D is always next to E but before G So 'ED' is effectively 1 letter => 7! Half of them will have 'G' after => 7! / 2

2. D after E 8! / 2

3. D before E and E before R 8!/3! since D,E,R can be arranged in 3! ways and we want only one of those ways.

4. D before E and E after R 8!/3 since we want 2 of the 6 arrangements of D,E,R

5. Vowels at even places

Choose even places: 4C3 Permute vowels: 3! Permute consonants: 5! So 4C3 * 3! * 5!

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