In how many ways can 14 different pizza toppings be chosen from 16 available toppings?
The correct answer and explanation is:
To find the number of ways to choose 14 different pizza toppings from 16 available toppings, we use combinations because the order of choosing toppings does not matter.
Step 1: Understand the problem type
- You have 16 different toppings.
- You want to select 14 toppings.
- The order in which you select the toppings doesn’t matter (choosing topping A then B is the same as choosing B then A).
- This is a combination problem, not a permutation problem.
Step 2: Use the combination formula
The formula for combinations is: (nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n-r)!}
where
- nn = total number of items (16 toppings),
- rr = number of items to choose (14 toppings),
- !! denotes factorial (the product of all positive integers up to that number).
Step 3: Calculate the number of ways
(1614)=16!14!×(16−14)!=16!14!×2!\binom{16}{14} = \frac{16!}{14! \times (16 – 14)!} = \frac{16!}{14! \times 2!}
Since 2!=22! = 2, and 16!/14!=16×15×14!/14!=16×1516! / 14! = 16 \times 15 \times 14! / 14! = 16 \times 15, we get: (1614)=16×152=2402=120\binom{16}{14} = \frac{16 \times 15}{2} = \frac{240}{2} = 120
Final answer:
120\boxed{120}
There are 120 different ways to choose 14 toppings out of 16.
Explanation in detail:
When choosing toppings where order does not matter, combinations help us avoid counting duplicates caused by different orders of the same selection. For example, picking toppings A and B is the same as picking B and A.
Choosing 14 toppings from 16 is the same as excluding 2 toppings from 16 (because 16 – 14 = 2). Hence, the number of ways to choose 14 toppings is also equal to the number of ways to choose which 2 toppings to leave out. This symmetry is why (1614)=(162)\binom{16}{14} = \binom{16}{2}.
Factorials grow very fast, but using factorial properties simplifies calculations. For example, 16!/14!=16×1516! / 14! = 16 \times 15 because all the terms from 14 down to 1 cancel out.
Thus, this approach efficiently gives the total number of ways to pick 14 toppings from 16, and the answer is 120 possible unique combinations.