what-is-the-coefficient-of-x-3-in-the-expansion-of-1-x-x-2-x-3-x-4-x-5-6-

Question Number 8217 by tawakalitu last updated on 02/Oct/16

what is the coefficient of x^{3} in the expansion

of {(1 + x + x^{2} + x^{3} + x^{4} + x^{5})}^{6}

Commented by 123456 last updated on 04/Oct/16

1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6

1 + 1 + 1 = 1 + 2 = 3

3 : 6

1 + 2 : 6 \cdot 5 = 30

1 + 1 + 1 : \frac{6 \cdot 5 \cdot 4}{3!} = \frac{6 \cdot 5 \cdot 4}{3 \cdot 2} = 5 \cdot 4 = 20

6 + 30 + 20 = 56

Commented by sandy_suhendra last updated on 04/Oct/16

this is so simple, but can you more explain it ?

Commented by 123456 last updated on 04/Oct/16

we have

{(1 + x + x^{2} + x^{3} + x^{4} + x^{5})}^{6} =

(1 + \cdot \cdot \cdot + x^{5}) \dots (1 + \cdot \cdot \cdot + x^{5})

recal x^{3} = x^{2} x = x x x

we have to pick one factor from each

(), if we pick x^{3}, all other will be 1

we have 6 () so 6 ways

if we pick x^{2}, we need to pick a x and

all other will be 1, so

6 () \times 5 () = 30 ways

if we pick x, we pick x^{2} (above case)

or x and x, so

6 () \times 5 () \times 4 () = “ 120 {ways}^{″}

however

x \times x \times x = x \times x \times x (3! = 6 ways of perm x)

so

\frac{120}{6} = 20 ways

since all coef are 1 the coef is the sum

of number of way

6 + 30 + 20 = 56

Commented by sandy_suhendra last updated on 04/Oct/16

great! {thank}^{'} s for your answer

Commented by tawakalitu last updated on 04/Oct/16

Thanks so much . i really appreciate .

Answered by sandy_suhendra last updated on 03/Oct/16

{[(1 + x + x^{2}) + x^{3} (1 + x + x^{2})]}^{6}

= {[(x^{3} + 1) (x^{2} + x + 1)]}^{6}

= {(x^{3} + 1)}^{6} {(x^{2} + x + 1)}^{6}

{(x^{3} + 1)}^{6} = {(x^{3})}^{6} + 6 {(x^{3})}^{5} + \dots + {6 x}^{3} + 1

{[x^{2} + (x + 1)]}^{6}

= 1 {(x^{2})}^{6} + 6 {(x^{2})}^{5} (x + 1) + \dots + {6 x}^{2} {(x + 1)}^{5} + 1 {(x + 1)}^{6}

{6 x}^{2} {(x + 1)}^{5} = {6 x}^{2} (x^{5} + {5 x}^{4} + \dots + 5 x + 1) \Rightarrow {30 x}^{3}

{(x + 1)}^{6} = x^{6} + {6 x}^{5} + \dots + {20 x}^{3} + \dots + 1

the term of x^{3} comes from \Rightarrow {6 x}^{3} + {30 x}^{3} + {20 x}^{3} = {56 x}^{3}

so the coefficient of x^{3} is 56

Commented by tawakalitu last updated on 03/Oct/16

Thanks so much .