Permutations and Combinations: Question 6

Worked solution

Step 1: Identify the three independent choices

The customer makes three separate decisions: which starter, which main course, and which dessert. Each choice is made from a different set of options, and every combination of starter, main and dessert is a distinct meal deal. Choosing pizza-then-fish is a different meal deal from choosing pizza-then-chicken.

Step 2: Apply the multiplication principle

Because the three choices are made independently, the total number of meal deals is the product of the numbers of options at each stage:

• Starter: $4$ ways
• Main course: $6$ ways
• Dessert: $3$ ways

$4\times6\times3 = 72.$

Step 3: Check by building up in stages

First combine starter and main course: there are $4\times6=24$ starter-and-main combinations. Each of these $24$ combinations can then be paired with any of the $3$ desserts:

$24\times3 = 72.$

Both methods agree: $72$.

Why the other options are wrong

• A ($13$): this comes from adding $4+6+3$ instead of multiplying, but each meal deal needs one choice from each category, not just one choice from any single category.
• B ($18$): this is $6\times3$, the number of main-and-dessert combinations only, it ignores the starter choice.
• C ($24$): this is $4\times6$, the number of starter-and-main combinations only, it ignores the dessert choice.

Final answer

• Number of different meal deals $= \boxed{72}$, option D.