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Question: Find the coefficient of \({a^2}{b^3}{c^4}d\) in the expansion of \({\left( {a - b - c + d} \right)^{...

Find the coefficient of a2b3c4d{a^2}{b^3}{c^4}d in the expansion of (a−b−c+d)10{\left( {a - b - c + d} \right)^{10}}.

Explanation

Solution

Start by grouping the variables inside the expression , then try to find out the highest value of r to be used while expanding the expression . Use the same procedure for further expansion in order to get the desired powers of a, b ,c and d . Separate the coefficient along with the sign obtained and solve it in order to find the required value.

Complete step-by-step answer:
Given,
(a−b−c+d)10{\left( {a - b - c + d} \right)^{10}}
Let us assume x = (a – b) and y = (d – c)
So the given equation would become as below,
(x+y)10{(x + y)^{10}}
Now, That we need to find the coefficients of a2b3c4d{a^2}{b^3}{c^4}d
We know , sum of powers or superscripts of a, b, c and d is 5.
Which means we need to check for those coefficients only which make the sum of powers of a, b, c, d as 5.
So we will proceed further by using the binomial expansion formula for (p+q)n{(p + q)^n}
nCrprqn−r{}^n{C_r}{p^r}{q^{n - r}}where nCr=n!r!(n−r)!{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}
So, for (x+y)10{(x + y)^{10}}=
10C5x5y5{}^{10}{C_5}{x^5}{y^5}
Substituting the values of x and y, we get
⇒10C5(a−b)5(d−c)5\Rightarrow {}^{10}{C_5}{(a - b)^5}{(d - c)^5}
Now applying binomial expansion for both the terms again as per the required exponents of a, b , c, d ,we get
⇒10C5⋅[5C2⋅a2⋅(−b)3]⋅[5C1(−c)4⋅d]\Rightarrow {}^{10}{C_5} \cdot \left[ {{}^5{C_2} \cdot {a^2} \cdot {{( - b)}^3}} \right] \cdot \left[ {{}^5{C_1}{{( - c)}^4} \cdot d} \right]
Rearranging the terms we get
⇒−10C5⋅5C2⋅5C1⋅a2⋅b3⋅c4⋅d\Rightarrow - {}^{10}{C_5} \cdot {}^5{C_2} \cdot {}^5{C_1} \cdot {a^2} \cdot {b^3} \cdot {c^4} \cdot d
Now , the coefficients of a2b3c4d{a^2}{b^3}{c^4}d will be,
=−10C5.5C2.5C1 =−12600  = - {}^{10}{C_5}.{}^5{C_2}.{}^5{C_1} \\\ = - 12600 \\\
So , the coefficient of a2b3c4d{a^2}{b^3}{c^4}d is−12600 - 12600.

Note: Such similar questions can be solved by the same approach used as above. Attention must be given while expanding the binomial expression as any wrong input might lead to vague or incorrect answers. Also nCr=n!r!(n−r)!{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} value can never be negative, but the coefficients can be due to the nature of variable.