Find Coefficient of x^7 in (1+x+x^2+x^3...)^n: C(n+6,7)

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In summary, the conversation discusses the problem of finding the coefficient of x^7 in the expression (1 + x + x^2 + x^3 + ...)^n. The speaker initially thought the answer was C(n+6, 6), but the book states it is actually C(n+6, 7). The conversation then explores the explanation behind this discrepancy, with the conclusion that the formula for selection with repetition was misused.
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Problem: Find the coefficient of x^7 in (1 + x + x^2 + x^3 + ...)^n
I thought this should be C(n + 6, 6) since you have to distribute 7 x's between the n series with repetition. But my book says it is C(n + 6, 7). (C for combination) What is the explanation?
 
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Well, let's see what happens when n=0. The coefficient of x^7 is 0, and 6C6=1 (you answer) and 6C7=0 (their answer; it is a convention that it is zero).

You could prove it inductively, and it might (for once) give you an idea of what's going on (induction usually doesn't). Or just work it through for some small n to see what is happening.
 
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Ah, never mind, it was just a dumb error. I misused the formula for selection with repetition--mixed up my bins and my objects.
 

What is the formula for finding the coefficient of x^7 in (1+x+x^2+x^3...)^n?

The formula for finding the coefficient of x^7 in (1+x+x^2+x^3...)^n is C(n+6,7), where C represents the combination function.

What do the numbers n and 6 represent in the formula C(n+6,7)?

The number n represents the power to which the polynomial (1+x+x^2+x^3...) is raised, while 6 represents the degree of the polynomial minus 1.

How do you interpret the result of the formula C(n+6,7)?

The result of the formula C(n+6,7) represents the number of ways to choose 7 terms from a polynomial of degree n+6 with repetition allowed. In this case, it gives the coefficient of x^7 in the expanded form of the polynomial.

Is there a specific method for finding the coefficient of x^7 in (1+x+x^2+x^3...)^n?

Yes, the formula C(n+6,7) is the specific method for finding the coefficient of x^7 in (1+x+x^2+x^3...)^n. It is derived from the binomial theorem and the concept of combinations.

Can the formula C(n+6,7) be used for finding coefficients of other terms in the expanded form of (1+x+x^2+x^3...)^n?

Yes, the formula C(n+6,7) can be modified to find the coefficient of any term with a specific power in the expanded form of (1+x+x^2+x^3...)^n. It follows the same pattern of choosing a certain number of terms from a polynomial with repetition allowed.

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