Simple Permutation & Combination question

[mbolisetty]

Question: I have 2 slices of bread made of wheat and 2 slices of bread made of Rye (4 slices of bread in total). How many different types of sandwiches can be made ?

[If I were to answer this manually, answer would be 3 (1wheat+1wheat, 1rye+1wheat, 1rye+1rye). But I 'm not able to get the answer mathematically]

NOTE: I would like to know the solution in conventional Permutation&Combination way AND anagram approach.

[Ben Ku]

My recommendation on this problem is to simply list them. If you want to think about it in terms of Slots or Anagrams, you can read below, but it's really more complicated than it needs to be. The basic issue with the Slots and Anagrams approaches is that we need to subtract out the duplicate issues. Really, using these techniques are more applicable and helpful when we have more complex problems.

Hope that helps.



Slots Approach
If we use the slots approach, we know we need to select two slices of bread. For each slice, we have two choices: either wheat or rye. So we have (2)(2) = 4.

However, in this calculation, note that we're actually listing FOUR possibilities: WW, WR, RW, and RR. If it doesn't matter which is the bottom or top slice, we have a duplicate: WR is the same as RW. So we have to subtract it out. 4 - 1 = 3.

Anagram Approach
The Anagram Approach requires us to list out all four breads. Then we'll put a "Y" if we select it, and a "N" if we do not.

W1 W2 R1 R2
Y, Y, N, N

If we want to determine how many ways to arrange YYNN, it is 4!/(2!2!) = 6. If we list this out, we find that we have FOUR duplicates:
W1R1, W1R2, W2R1, W2R2 are all the same combination; three of them are extra. So 6 - 3 = 3.

[ps63739]

As Ben said, simple logic will solve this problem much quicker and without confusion than the two methods.

Just imagine you have two types of breads and how many different combinations can be possible for making a sandwiches

Wheat and Wheat, Wheat and Rye, Rye and Rye.

[Ben Ku]

Exactly.

[dhruv285]

Lets take this problem as a kind of a permutation problem.

We have to arrange ( as a sandwich ) 2 pieces of bread from 4 pieces of bread. The number of ways to do this will be 4P2 which equals 12. Now lets add the information that there are two pieces of wheat bread and two pieces of rye bread. So now since there is repetition the formulae will be (4P2)/(2!*2!). Each of the 2! Corresponds to the repetition.

Using the enhanced formulae we get the answer as 3. What do you think?

[RonPurewal]

Using the enhanced formulae we get the answer as 3. What do you think?

what do i personally think? well, i personally think it's ... unwise to apply such complex formulas to a situation as simple and limited as this one.
i mean, if you just list things, you can solve this thing in between 5 and 15 seconds.

on the other hand, yes, it seems that you understand these formulas well enough to extend them to problems on which they really are useful.