I need to know a combinatoric solution to this problem, with Generating functions in book, gives us 102.

It might be a very simple problem, but Im very confused with this. Would be very nice of you to give me some help. Im really grateful.

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The number of ways to arrange EEGI: $frac4!2!cdot1!cdot1! =12$The number of ways to arrange EEGN: $frac4!2!cdot1!cdot1! =12$The number of ways to arrange EEIN: $frac4!2!cdot1!cdot1! =12$The number of ways to arrange EENN: $frac4!2!cdot2! = 6$The number of ways to arrange EGIN: $frac4!1!cdot1!cdot1!cdot1!=24$The number of ways to arrange EGNN: $frac4!1!cdot1!cdot2! =12$The number of ways to arrange EINN: $frac4!1!cdot1!cdot2! =12$The number of ways to arrange GINN: $frac4!1!cdot1!cdot2! =12$
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answered Dec 6 "15 at 10:37
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barak manosbarak manos
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We can view the letters of the word ENGINE as a multiset with two E"s, two N"s, one G, and one I, that is $2 cdot E, 2 cdot I, 1 cdot G, 1 cdot I$. When we select four of these letters, we will either have four different letters, three different letters with one repeat, or two different letters with each repeated.

Case 1: Four different letters.

This is a permutation of the letters E, N, G, I. They can be permuted in $4!$ ways.

Case 2: Three different letters with one repeated.

There are two ways to select the repeated letter from $E, N$. We can choose the locations of the repeated letter in $inom42$ ways. There are three ways to fill the leftmost open spot with one of the other three letters and two ways to fill the last spot with one of the two letters that has not yet been selected. Hence, there are $$inom21inom42inom31inom21$$arrangements with three different letters in which one letter is repeated.

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Case 3: Two different letters with both repeated.

We permute two E"s and two N"s. There are $inom42$ ways to select the locations of the two E"s. There is one way to fill the remaining spots with the two N"s. Hence, the number of ways to arrange two different letters with both repeated is$$inom42$$

Thus, the total number of ways of arranging four letters of the word ENGINE is $$4! + inom21inom42inom31inom21 + inom42$$