# Thread: Coefficient of x^4 in the product of the expansions (1 + x)^5.(1 - x)^6 is?

1. ## Coefficient of x^4 in the product of the expansions (1 + x)^5.(1 - x)^6 is?

Coefficient of x^4 in the product of the expansions (1 + x)^5.(1 - x)^6 is -

A) 5C2
B) ∑(r = 0 to 4) (-1)^r 6Cr . 5C(4 - r)
C) - 5C3
D) 5C4

More than one options may be correct.

2. The general term of $\displaystyle (1+x)^5$ is $\displaystyle C_5^kx^k, \ 0\leq k\leq 5$

The general term of $\displaystyle (1-x)^6$ is $\displaystyle (-1)^lC_6^lx^l, \ 0\leq l\leq 6$

Then the genaral term of the entire expansion is $\displaystyle (-1)^lC_5^kC_6^lx^{k+l}, \ 0\leq k\leq 5, \ 0\leq l\leq 6$

So the condition is $\displaystyle k+l=4$.

Now take all the possibilities for k and l.

3. Originally Posted by fardeen_gen
Coefficient of x^4 in the product of the expansions (1 + x)^5.(1 - x)^6 is -

A) 5C2
B) ∑(r = 0 to 4) (-1)^r 6Cr . 5C(4 - r)
C) - 5C3
D) 5C4

More than one options may be correct.
Hint:

$\displaystyle {\left( {1 + x} \right)^5}{\left( {1 - x} \right)^6} = {\left( {1 + x} \right)^5}{\left( {1 - x} \right)^5}\left( {1 - x} \right) = {\left( {1 - {x^2}} \right)^5}\left( {1 - x} \right) = {\left( {{x^2} - 1} \right)^5}\left( {x - 1} \right)$

4. The general term of is

The general term of is

Then the genaral term of the entire expansion is

So the condition is .

Now take all the possibilities for k and l.
Correct me if I am wrong. 5C4 is the only possible option.

5. No.
You have the possibilities:
k=0, l=4
k=1, l=3
k=2, l=2
k=3, l=1
k=4, l=0

Then the coefficient is

$\displaystyle C_5^0C_6^4-C_5^1C_6^3+C_5^2C_6^2-C_5^3C_6^1+C_5^4C_6^0=\sum_{r=0}^4(-1)^rC_6^rC_5^{4-r}$

6. Members of the decomposition of the factor $\displaystyle {\left( {{x^2} - 1} \right)^5}$ have only even powers. Therefore, you just enough to consider this factor.

Your answer must be $\displaystyle C_5^3 = 10$ for $\displaystyle x^4$.

B), C) and D), right?

8. No, the correct answer is B.

9. $\displaystyle {\left( {{x^2} - 1} \right)^5} = \sum\limits_{k = 0}^5 {C_5^k{x^{2\left( {5 - k} \right)}}{{\left( { - 1} \right)}^k}} = C_5^0{x^{10}} - C_5^1{x^8} + C_5^2{x^6} - C_5^3{x^4} + C_5^4{x^2} - C_5^5{x^0}$

$\displaystyle {\left( {1 + x} \right)^5}{\left( {1 - x} \right)^6} = {\left( {{x^2} - 1} \right)^5}\left( {x - 1} \right) \Rightarrow C_5^3 = 10{\text{ for }}{x^4}$

Sorry, I corrected.

10. Originally Posted by DeMath
Only C)

$\displaystyle {\left( {{x^2} - 1} \right)^5} = \sum\limits_{k = 0}^5 {C_5^k{x^{2\left( {5 - k} \right)}}{{\left( { - 1} \right)}^k}} = C_5^0{x^{10}} - C_5^1{x^8} + C_5^2{x^6} - C_5^3{x^4} + C_5^4{x^2} - C_5^5{x^0}$

$\displaystyle (1+x)^5(1-x)^6\neq (x^2-1)^5$ !!!

11. red_dog, A) is equivalent to B). So A) is also correct, right?

And aren't we supposed to take the individual possibility of k = 4 and l = 0, which gives 5C4(option D)?