Question
Question: What will be the coefficients of \({a^8}{b^4}{c^9}{d^9}\) in \({\left( {abc + abd + acd + bcd} \righ...
What will be the coefficients of a8b4c9d9 in (abc+abd+acd+bcd)10.
Solution
Hint: In this question use the direct formula for any general term for equation in form of (abc+abd+acd+bcd)nwhich is x!.y!.z!.q!n!(abc)x(abd)y(acd)z(bcd)q=x!.y!.z!.q!n!a(x+y+z)b(x+y+q)c(x+z+q)d(y+z+q).The direct power and coefficients comparison will get to the answer.
Complete step-by-step answer:
As we know the general term of (abc+abd+acd+bcd)10 is
x!.y!.z!.q!10!(abc)x(abd)y(acd)z(bcd)q=x!.y!.z!.q!10!a(x+y+z)b(x+y+q)c(x+z+q)d(y+z+q)
Now we need the coefficient of a8b4c9d9
So on comparing this with above equation we have,
⇒x+y+z=8................... (1)
⇒x+y+q=4................... (2)
⇒x+z+q=9................... (3)
⇒y+z+q=9.................. (4)
Now add all the four equation we have,
⇒3x+3y+3z+3q=8+4+9+9=30
Now divide by 3 we have,
⇒x+y+z+q=10 ............................. (5)
Now subtract equation (1), (2), (3) and (4) from equation (5) respectively we have,
⇒x+y+z+q−x−y−z=10−8
⇒q=2
And
⇒x+y+z+q−x−y−q=10−4
⇒z=6
And
⇒x+y+z+q−x−z−q=10−9
⇒y=1
And
⇒x+y+z+q−y−z−q=10−9
⇒x=1
Therefore the coefficient of a8b4c9d9 is
⇒1!.1!.6!.2!10!a8b4c9d9
Now simplify the above equation we have,
⇒1×1×6!×2×110×9×8×7×6!a8b4c9d9
⇒210×9×8×7a8b4c9d9
⇒2520a8b4c9d9
So the required coefficient of a8b4c9d9 in (abc+abd+acd+bcd)10 is 2520.
So this is the required answer.
Note: Such types of questions are direct formula based and it is always advised to remember these direct formulas. It’s not a binomial expansion so we need not to be confused between these two concepts. Both are different to each other, any general term in the binomial expansion of (x+y)nis its (n−r+2)th term.