Discrete Structures207910 marksNumerical: combinations and induction proofmathematical Induction
A group of 8 scientist is composed of 5 chemist and 3 biologist. In how many ways can a committee of 5 be formed that has 3 chemist and 2 biologist? Using mathematical induction prove that 13 + 23 + 33 + ... + n3 = (n2(n+1)2/4) for n geq 1.
A group of 8 scientist is composed of 5 chemist and 3 biologist. In how many ways can a committee of 5 be formed that has 3 chemist and 2 biologist? Using mathematical induction prove that $1^3 + 2^3 + 3^3 + ... + n^3 = (n^2(n+1)^2/4)$ for $n \geq 1$. [10]
- Total scientists = 8 (5 chemists + 3 biologists) - Committee required = 5 members: exactly 3 chemists and 2 biologists Select 3 chemists from 5: $$C(5,3) = \frac{5!}{3!,2!} = \frac{5\times 4}{2\times 1} = 10$$ Select 2 biologists from 3: $$C(3,2) = \frac{3!}{2!,1!} = 3$$ By the multiplication...