An organizing committee is to be formed by selecting 4 men and 3 women from a group of 8 men and 6 women. If two specific men refuse to serve on the committee together, and two specific women also refuse to serve on the committee together, then the number of ways the committee can be formed so that neither of these conflicts occurs is _____.

Step-by-step solution

Total number of ways to select 4 men from 8 and 3 women from 6 without any restrictions is ⁸C₄ ⁶C₃ = 70 20 = 1400 Let n(A) be the number of ways the two specific men are selected together. We must choose 2 more men from the remaining 6 , and 3 women from 6 . n(A) = ⁶C₂ ⁶C₃ = 15 20 = 300 Let n(B) be the number of ways the two specific women are selected together. We must choose 4 men from 8 , and 1 more woman from the remaining 4 . n(B) = ⁸C₄ ⁴C₁ = 70 4 = 280 Let n(A B) be the number of ways both pairs are selected