Exhaustively Yours Problem
How many different 9-letter words can be made by simply removing two letters from "MISSISSIPPI"? For instance, SSISSIPPI, MISSSIPPI, are MISSISSPP 9-letter words made by removing two letters from MISSISSIPPI.
Note: Rearrangement of letters is NOT allowed.
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1 solution
I think the ans is 31.
There are of course ( 1 1 2 ) combinations. But
(i) if we remove 'M' and first 'S', we get the same combination as second 'S' and so on. Hence, for 'M', we get 3 such 'repetitions', for 'I' too 3, for 'S' 2 and so on. M − 3 , I − 3 , S − 2 , I − 2 , S − 1 , I − 1 , here we have only counted S and I once only.
(ii) Now, S is repeated 2 times consecutively, so removing first S and the fore-coming alphabets and removing second S and the fore-coming alphabets will produce the same combinations(except the first S with the second S).
Hence, we may also eliminate all the combinations of the second S and so on. This can be written as-
2 n d S − 7 , 4 t h S − 4 , 2 n d P − 1
Adding all the eliminations and then subtracting it from 55, we get
5 5 − ( 1 + 1 + 2 + 2 + 3 + 3 + 7 + 4 + 1 ) = 5 5 − 2 4 = 3 1
Please tell me the one selections which I missed.
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@Rajen Kapur , care to tell my mistake! @Calvin Lin , Sir, please care to check this problem!
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@Calvin I appreciate your systematical thinking. The only thing that is missing in your detail is that when you remove middle I of MISSISSIPPI you are counting one next S to be giving a new word. That is the mistake you are making as that is not a new word, having been already obtained when this I was removed with the previous S. For illustration let us apply your analysis to ISSISS. Out of 15 combinations, deductions for I - 2, S - 1, I - 1 when counting S once only. Now take second S - 3. Actually, there are SISS, SSSS, SSIS, IISS, ISSS, ISIS, ISSI only 7 words in all.
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@Rajen Kapur – Oh, yes! Thanks for clearing my doubt! Nice Problem! But unfortunately I missed it by mere inches!
shouldn't the ans be 11C2=55??