Question

Number of ways in which $3$ boys and $3$ girls (all are of different heights) can be arranged in a line so that boys of as well as girls among themselves are in decreasing order of height (from left to right), is:

• A. $1$
• B. $6!$
• C. $20$
• D. None of these

Answer 1

Answer: (C)

$20$

• Let's assume we have B1, B2, B3 (in order of descending height) and G1, G2, G3 (in order of descending height).
  

• We want to figure the total patterns of boys and girls we can form. For example, BBGBGG is one pattern. Once we have a pattern, there is one way to slot each of the children into the slots. Specifically, for BBGBGG, we would arrange them B1, B2, G1, B3, G2, G3.
  

• Okay, so the problem reduces to finding the patterns of 3 boys within 6 positions (or equivalently 3 girls within 6 positions).
  

• That's just "6 choose 3" --> 6C3
  
• 6C3 = 6 x 5 x 4 / 3!
  

= 120 / 6
= 20 ways

• Another approach
  

Take the 6 children and count the total possible arrangements --> 6! = 720 ways
Of these 1 out of 3! of them will have the boys in the correct order (1/6)
Of these 1 out of 3! of them will have the girls in the correct order (1/6)
So we need to divide by (3! x 3!) or 36
6! /(3!) (3!) = 720/36 = 20 ways