There are four scoops of ice cream on one cone they are four different flavors how many ways can they be placed on the cone?

Answer 1

The answer is 24.

.

The key to this answer is in the wording of the question - there are four different flavors of ice cream on the cone - meaning no flavor can appear more than once on the cone.

.

This particular problem is called a Permutation, and is mathematically expressed as nPr, meaning how many ways can you order n things when selecting r things at a time. The calculation of a permutation is:

nPr = n! / ((n - r)!)

where '!' is the factorial function.

.

In this particular question, both n and r equal 4 - there are 4 things, and you are selecting 4 of them at a time. so, the expression equates to:

4P4 = 4! / ((4 - 4)!) = 24 / 1 = 24

.

Or, we can arrange the four things in groups of four, and see how many permutations there are. Let's reptesent the four ice cream flavors with the letters a, b, c, and d. When we look at how many ways we can order the four flavors when selecting four at a time, we get the following list of permutations:

1 a,b,c,d

2 a,b,d,c

3 a,c,b,d

4 a,c,d,b

5 a,d,b,c

6 a,d,c,b

7 b,a,c,d

8 b,a,d,c

9 b,c,a,d

10 b,c,d,a

11 b,d,a,c

12 b,d,c,a

13 c,a,b,d

14 c,a,d,b

15 c,b,a,d

16 c,b,d,a

17 c,d,a,b

18 C,d,b,a

19 d,a,b,c

20 d,a,c,b

21 d,b,a,c

22 d,b,c,a

23 d,c,a,b

24 d,c,b,a

As we can see, there are 24 permutations.

.

If we were allowed duplicates, that is, any arrangement of flavors, where a flavor could occur none, once, twice, three times, or even four times, then the answer would be 256. In this case, the mathematical expression is

.

S = an

.

where 'a' is the set of things chosen (ice cream flavors) and 'n' is the number of times it is selected. In this case, both n and a are 4, so the equation works out as follows.

.

S = 44 = 256.

.

If we were to list the possible arrangements, it would be as follows.

1 a,a,a,a

2 a,a,a,b

3 a,a,a,c

4 a,a,a,d

5 a,a,b,a

6 a,a,b,b

7 a,a,b,c

8 a,a,b,d

...

252 d,d,c,d

253 d,d,d,a

254 d,d,d,b

255 d,d,d,c

256 d,d,d,d

Answer 2

12

* * * * *

That is not the correct answer, however you look at it. And, unfortunately, the question is not clear enough.

• Can you repeat the flavours?
• Does the order matter?

So, for example,

No repeats, order matters: 4P3 = 4*3*2 = 24 ways.

No repeats, order does not matter : 4C3 = 4*3*2/(3*2*1) = 4 ways.

Answer 3

16 since you mulitiply 4 times 4 to receive the answer

Answer 4

as manny ways as you want

Answer 5

6 ways

Answer 6

The answer is 4*3*2*1 which is 24

Answer 7

256