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Answer :
We start with the fact that order does not matter when choosing ice cream scoops or toppings.
(a) For the bowl of 4 scoops where each scoop is a different flavor from 22 available flavors, we calculate the number of ways using the combination formula:
[tex]$$
\binom{22}{4} = \frac{22!}{4!(22-4)!} = 7315.
$$[/tex]
(b) Next, for the toppings, you choose 4 out of 9 available toppings. Again, order does not matter, so the number of ways is:
[tex]$$
\binom{9}{4} = \frac{9!}{4!(9-4)!} = 126.
$$[/tex]
Since the ice cream scoops and the toppings are chosen independently, the total number of ways to select 4 scoops and 4 toppings is the product of the two quantities:
[tex]$$
7315 \times 126 = 921690.
$$[/tex]
Thus, the answers are:
- (a) There are [tex]$7315$[/tex] ways to select 4 different ice cream flavors.
- (b) There are [tex]$921690$[/tex] ways to select a bowl of 4 scoops and 4 toppings.
(a) For the bowl of 4 scoops where each scoop is a different flavor from 22 available flavors, we calculate the number of ways using the combination formula:
[tex]$$
\binom{22}{4} = \frac{22!}{4!(22-4)!} = 7315.
$$[/tex]
(b) Next, for the toppings, you choose 4 out of 9 available toppings. Again, order does not matter, so the number of ways is:
[tex]$$
\binom{9}{4} = \frac{9!}{4!(9-4)!} = 126.
$$[/tex]
Since the ice cream scoops and the toppings are chosen independently, the total number of ways to select 4 scoops and 4 toppings is the product of the two quantities:
[tex]$$
7315 \times 126 = 921690.
$$[/tex]
Thus, the answers are:
- (a) There are [tex]$7315$[/tex] ways to select 4 different ice cream flavors.
- (b) There are [tex]$921690$[/tex] ways to select a bowl of 4 scoops and 4 toppings.
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