Multinomial Expansion: Combinations of Letters in MISSISSIPPI Taken 5 at a Time

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To find the number of combinations of the letters in "MISSISSIPPI" taken 5 at a time, the approach involves using multinomial expansions. The key expression to consider is (1+x)(1+x+x^2)(1+x+x^2+x^3+x^4)^2, where the goal is to determine the coefficient of x^5. While expanding the expression is one method, it can be tedious, leading to inquiries about potential shortcuts. The discussion highlights the challenge of managing terms with exponents limited to 5. Ultimately, the focus remains on finding an efficient way to calculate the desired combinations.
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Homework Statement


Find the number of combinations of the letters occurring in the word "MISSISSIPPI" taken 5 at a time.


The Attempt at a Solution



Using multinomials,
The no. of combinations is the coeff. of x5 in the expression
(1+x)(1+x+x2)(1+x+x2+x3+x4)2

I don't know how to proceed after this.
Any help appreciated.
 
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While tedious, I would expand. In the process, you only ever need to keep terms with exponent less than or equal to 5. (Why?)
 
Thanks alot!...but is there any shortcut method for this?
 
I don't know one. Perhaps.
 

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