[Guest]

An ice cream parlor in a remote island has 14 different flavors of ice creams: A, B, C, D, E, F, G, H, I, J, K, L, M, N.

John, who is a visitor, has a choice of 4 different types of cones: W, X, Y, Z.

(a) How many distinct 3-scoop cones can he order?

(b) How many distinct 4-scoop cones can he order?

Notes:

(i) A given 3-scoop cone may contain one or more flavors. For example, in a given cone A-A-B (top to bottom) is a valid combination.

(ii) All permutations of a given 3-scoop cone combination are deemed the same. For example, in a given cone, A-B-C and A-C-B are treated as the same combination.

(iii) The same logic in terms of (i) and (ii) is applicable in case of 4-scoop cones.

[Guest]

There are 2240 possible combinations for a 3-scoop cone and 3384 combinations for a 4-scoop cone.

3-Scoops:

When all the scoops are the same flavor, then there are 14 possibilities.

When 2 scoops are the same flavor, then there are 14 X 13 = 182 possibilities

When all scoops are different, then take the sum i=1 to 12 [sum j=1 to i [j]] to get 364 possibilities.

This makes a total of 14 + 182 + 364 = 560 choices per cone.

Multiply by the number of cones, 560 * 4 = 2240 total 3-scoop combinations.

4-Scoops is very similar:

When all the scoops are the same flavor, then there are 14 possibilities.

When 3 scoops are the same flavor, then there are 14 X 13 = 182 possibilities

When 2 scoops are the same flavor, then take the sum i=1 to 12 [sum j=1 to i [j]] to get 364 possibilities.

When all scoops are different, then take the sum i=1 to 11 [sum j=1 to i [j]] to get 286 possibilities.

This makes a total of 14 + 182 + 364 + 286 = 846 choices per cone.

Multiply by the number of cones, 846 * 4 = 3384 total 4-scoop combinations.

[Guest]

ABC = (14*13*12)/6 = 364

AAB = (14* 1*13)/1 = 182

AAA = (14* 1* 1)/1 = 14

---

560 * 4 = 2240

ABCD = (14*13*12*11)/24 = 1001

AABB = (14* 1*13* 1)/2 = 91

AABC = (14* 1*13*12)/2 = 1092

AAAB = (14* 1* 1*13)/1 = 182

AAAA = (14* 1* 1* 1)/1 = 14

----

2380 * 4 = 9520

[Guest]

kanth_sri, you should divide by 6 instead of 2 for AABC.

My updated answer for 4 scoops is:

ABCD = (14*13*12*11)/24 = 1001

AABB = (14* 1*13* 1)/2 = 91

AABC = (14* 1*13*12)/6 = 364

AAAB = (14* 1* 1*13)/1 = 182

AAAA = (14* 1* 1* 1)/1 = 14

-------

1652 * 4 = 6608

[Guest]

Jobe17, I think it should be divided by 2 and not 6.

kanth_sri, you should divide by 6 instead of 2 for AABC.

[Guest]

An ice cream parlor in a remote island has 14 different flavors of ice creams: A, B, C, D, E, F, G, H, I, J, K, L, M, N.

John, who is a visitor, has a choice of 4 different types of cones: W, X, Y, Z.

(a) How many distinct 3-scoop cones can he order?

(b) How many distinct 4-scoop cones can he order?

Notes:

(i) A given 3-scoop cone may contain one or more flavors. For example, in a given cone A-A-B (top to bottom) is a valid combination.

(ii) All permutations of a given 3-scoop cone combination are deemed the same. For example, in a given cone, A-B-C and A-C-B are treated as the same combination.

(iii) The same logic in terms of (i) and (ii) is applicable in case of 4-scoop cones.

[Guest]

One each, assuming he has two hands.