MATHCOUNTS RULES
On the refrigerator, MATHCOUNTS is spelled out with 10 magnets, one letter per magnet. Two vowels and three consonants fall off and are put away in a bag. If the Ts are indistinguishable, how many distinct possible collections of letters could be in the bag?
Source:Mathcounts Nationals
The answer is 75.
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1 solution
Now to the refrigerator letters: there are 3 vowels and 7 consonants, two of which (the Ts) are the same. There are 3 ways to combine two vowels in the bag (depending on which one is left on the refrigerator).
If there are two Ts in the bag, the third consonant can be any of the other 5. If there are not two Ts, the consonants in the bag can be any three of six possibilities (maybe including one T, maybe not). So the total number of consonant combinations is 5C1 + 6C3 = 5 + [6! / (3! 3!)] = 5 + 20 = 25 and the total number of possible collections of letters in the bag is the product of the vowel and consonant combinations: 3C2 * 25 = 75