Question

In how many ways can one wrap 3 KitKat, 2 FiveStar and 3 BarOne chocolates in a gift pack containing exactly three chocolates, if at least one KitKat must be included? (Treat all chocolates as distinct.)

• A. 3
• B. 10
• C. 46
• D. 56

Hint:

“At least one of a type” with distinct items is fastest by complement: count all, subtract the selections with none of that type. A case split by the exact count is a good cross-check.

Answer 1

The Correct Option is C

We are choosing 3 distinct chocolates from a set of 8 (\(3+2+3\)), subject to “at least one KitKat.” Method 1 (Complement).
Total unrestricted selections: \[ \binom{8}{3}=56. \] Selections with \emph{no} KitKat use only the \(5\) non-KitKats (2 FiveStar + 3 BarOne): \[ \binom{5}{3}=10. \] Therefore valid selections: \[ 56-10=46. \] Method 2 (Case split by \# of KitKats).
\[ \begin{aligned} \text{Exactly 1 KitKat: }& \binom{3}{1}\binom{5}{2}=3\cdot 10=30,
\text{Exactly 2 KitKats: }& \binom{3}{2}\binom{5}{1}=3\cdot 5=15,
\text{Exactly 3 KitKats: }& \binom{3}{3}\binom{5}{0}=1\cdot 1=1. \end{aligned} \] Sum \(=30+15+1=46\) (agrees with Method 1). \[ \boxed{46} \]