Question

Evaluate the following: (a) 5​C2​ (b) (73​) Written Response: Show your work for full marks. (4) 10. Solve for n, and state the restrictions. 2⋅(n+2)!/ (n+1)!​=12

Answer 1

To evaluate 5C2, we use the combination formula, yielding a result of 10. To solve 2·(n+2)!/ (n+1)!=12 for n, we simplify the equation to find that n=4, with the restriction that n must be a non-negative integer.

For evaluating (a) 5​C2, we use the combination formula ℒCₙ = n! / [k!(n-k)!]. Here, n is 5 and k is 2, so:

5C2 = 5! / [2!(5-2)!] = 120 / [2*6] = 10.

For solving (b) 2·(n+2)!/ (n+1)!​=12, we factor out (n+1)! from both the numerator and the denominator:

2·(n+2)(n+1)! / (n+1)! = 12

(n+2) = 6 → n = 4.

Restrictions: The solution must be a non-negative integer, since factorials are defined for non-negative integers only.