GMAT Math

1) The Natural Woman, a women's health food store, offers its own blends of trail mix. If the store uses 4 different ingredients, how many bins will it need to hold every possible blend, assuming that each blend must have at least two ingredients? (Also, assume that each bin can hold one and only one blend)

I am trying to solve the first question by using the following method; however it is not giving me the same reply.
Since this is an " at least' question: the total number of arrangements less the undesired outcomes should give us the desired outcome. IN this case, the total number of arrangements should be 4!=4*3*2*1=24 and the undesired outcome is to have 1 ingredient and zero ingredient. Having 1 ingredient is equivalent to 4!/3!=4 and having zero ingredients is equivalent to 4!/4!=1 ==> the total undesired outcomes=4+1=5 less the total arrangements =24, which results in the number of bins=24-5=19 . However the answer that you are giving is 11. Where did I go wrong in my analysis?? Please advise. Thanks!

Hi,

First off, this doesn't appear to be a permutation question (you used the word "arrangement") since you're blending things together (and when you blend things together, it doesn't matter what goes into the mix first). You need to use the combination formula to solve this. There aren't 24 ways to do things either, but there are:

1 item: 4 ways
2 items: 4!/(2!)(2!) = 6 ways
3 items: 4!/(3!)(1!) = 4 ways
4 items: 1 way

Since the question asks about blending 2 or more ingredients, there are 11 blends that could exist.

-Josh