Browse · MATH
Printjmc
counting and probability senior
Problem
On the refrigerator, MATHEMATICS is spelled out with magnets, one letter per magnet. Two vowels and four consonants fall off and are put away in a bag. If the T's, M's, and A's are indistinguishable, how many distinct possible collections of letters could be put in the bag?
Solution
We count the number of ways to choose the vowels and the consonants separately. There are four vowels, of which two are As. If there are no As, then we must choose both the remaining vowels, so there is choice; if there is one A, then we can choose the remaining vowel in ways; and if there are two As, then there are no vowels left to choose, so there is choice. This makes distinct pairs of vowels.
There are seven consonants, of which two are Ts and of which two are Ms. Since we must choose four consonants, we must use at least one of the Ts and Ms.
If we use one T and no Ms, we have only choice (use the three remaining consonants); the same is true if we use one M and no Ts. If we use both Ts and no Ms, there are choices for the two remaining consonants; the same is true if we use both Ms and no Ts, or if we use one T and one M. If we use both Ts and one M, there are choices for the single remaining consonant; the same is true if we use both Ms and one T. Finally, if we use both Ts and both Ms, there are no more letters left to choose, so we get more choice.
In total, we have distinct collections of consonants.
Therefore, the number of distinct collections of letters is
There are seven consonants, of which two are Ts and of which two are Ms. Since we must choose four consonants, we must use at least one of the Ts and Ms.
If we use one T and no Ms, we have only choice (use the three remaining consonants); the same is true if we use one M and no Ts. If we use both Ts and no Ms, there are choices for the two remaining consonants; the same is true if we use both Ms and no Ts, or if we use one T and one M. If we use both Ts and one M, there are choices for the single remaining consonant; the same is true if we use both Ms and one T. Finally, if we use both Ts and both Ms, there are no more letters left to choose, so we get more choice.
In total, we have distinct collections of consonants.
Therefore, the number of distinct collections of letters is
Final answer
72