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Find the coefficients of xnyn in the expansion of the following expression,
[(1+x)(1+y)(x + y)]n.

Answer
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Hint: In order to find the coefficients of xnyn from the given expansion, we use the formulae of binomial expansions of each of the individual terms from the given expansion, i.e., (1+x)n, (1+y)n and (x+y)n respectively to figure out the coefficients.

Complete step-by-step solution:
Given Data,
Expansion - [(1+x)(1+y)(x + y)]n
Coefficients of xnyn
From the concept of binomial expansion, let us use the formula of binomial expansion of each term from the given equation,
We know the formula of the term (1+x)n is given by
(1+x)n=C0+C1x + C2x2+.......+Cnxn
We know the formula of the term (1+y)n is given by
(1+y)n=C0yn+C1yn - 1 + C2yn - 2+.......+Cn
We know the formula of the term (x+y)n is given by
(x+y)n=C0xn+C1xn - 1y + C2xn - 2y2+.......+Cnyn
If we multiply all these binomial expansions of terms, each vertical row will be the coefficient of the term xnyn, i.e.
The first terms of all the equations, gives a variable value of xnyn with a coefficient ofC03.
The coefficient of the first term of the equations is C0×C0yn×C0xn=C03xnynisC03.
When we multiply the expansions of (1+x)n,(1+y)n and (x+y)n, the first, second and so on nth terms of the equations will contain the same variable term which is xnyn and a coefficient C03,C13…….Cn3 for the first, second and so on the nth term of the product equation of all three expansions.
Similarly the coefficient of xnyn for the last term of the product of binomial expansions of each term will also be Cn3.
Therefore the coefficients of xnyn in the expansion [(1+x)(1+y)(x + y)]n is given as:
C03+C13+C23+......+Cn3.

Note: In order to solve this type of problem the key is to know the concept of binomial expansion and the respective formulae of binomial expansion to the power of n for each of the term in the given expansion. The key step in solving this problem is identifying product of each of the respective term in each binomial expansion results in the forming term xnyn. So the sum of all the individual coefficients gives us the coefficient of xnyn for the given expansion.

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