5 ingredients, need 2 to make something - how many combos?

Question

If you have 5 ingredients, and need 2 to make something -how many combinations could you have?

Is it 5x4=20?
If you didnt need 2 ingredients, would it be 5x4x3x2=120?

(ignoring the notion that the amount of each ingredient could be altered for a new combination)

Answer 1

The first answer is correct, namely 5C2 = 10. Nixonbot's answer is not correct (namely 20) because it double counts everything. For instance, if the ingredients were labelled a,b,c,d,e, it would count {a,b} as different from {b, a}. But it does not matter in which order you choose your ingredients, so you have to divide by 2.

In general if you have n objects and want to choose k of these objects and you DON'T care about order, there will be

n!/[(n-k)!k!]

ways of choosing.

If you DO care about order, then there will be

n!/(n-k)!

ways of choosing.

On a scientific calculator, the former is found using the nCk button, the latter with the nPk button.

To answer your second question, if you didn't need 2 ingredients, then you need the other 3 ingredients (presumably this what you mean?) so it is the same as computing 5C3 = 5!/[(5-3)!3!] = 10.

Answer 2

You would be correct because you have a choice of 5 ingredients for the first choice, and then 4 left for the second one. 5*4=20

Answer 3

1+2
1+3
1+4
1+5
2+3
2+4
2+5
3+4
3+5
4+5

A total of 10 possibilities without duplicating.

Answer 4

only 10 combos... 5 times 4 is the right answer if you want to put them in order so to get the right answer you would have to take 5 times 4 then divide by 2 you're on the right track but the flaw is that combo x/y is the same as combo y/x

Answer 5

Obviously the correct answer is 5!/(3!2!) = 10

To nixonbot - although your logic is correct, you forgot thet choosing A first and then B is equivalent to choosing B first and then A (the order does not matter)!

Answer 6

if just using one of the ingredients alone counts the answer is 15 if not then its 10

Answer 7

Nope, only 10

Answer 8

5C2 = 5!/(3!2!) = 10

Answer 9

THE ANSWER IS TEN. LIKE THIS....INGREDIENTS=A B C D E
COMBINATIONS...AB,AC,AD,AE,BC,
BD,BE,CD,CE,DE.=10
AS WAS EXPLAINED AB=BA... etc.

Answer 10

You're correct.