How many terms does the binomial expansion of (x^2+7y) ^9

x

18

+

63

x

16

y

+

1764

x

14

y

2

+

28812

x

12

y

3

+

302526

x

10

y

4

+

2117682

x

8

y

5

+

9882516

x

6

y

6

+

29647548

x

4

y

7

+

51883209

x

2

y

8

+

40353607

y

9

Step-by-step explanation:

(

a

+

b

)

n

=

n

∑

k

=

0

n

C

k

⋅

(

a

n

-

k

b

k

)

.

9

∑

k

=

0

9

!

(

9

-

k

)

!

k

!

⋅

(

x

2

)

9

-

k

⋅

(

7

y

)

k

Expand the summation.

9

!

(

9

-

0

)

!

0

!

⋅

(

x

2

)

9

-

0

⋅

(

7

y

)

0

+

9

!

(

9

-

1

)

!

1

!

⋅

(

x

2

)

9

-

1

⋅

(

7

y

)

+

...

+

9

!

(

9

-

9

)

!

9

!

⋅

(

x

2

)

9

-

9

⋅

(

7

y

)

9

Simplify the exponents for each term of the expansion.

1

⋅

(

x

2

)

9

⋅

(

7

y

)

0

+

9

⋅

(

x

2

)

8

⋅

(

7

y

)

+

...

+

1

⋅

(

x

2

)

0

⋅

(

7

y

)

9

Simplify the polynomial result.

x

18

+

63

x

16

y

+

1764

x

14

y

2

+

28812

x

12

y

3

+

302526

x

10

y

4

+

2117682

x

8

y

5

+

9882516

x

6

y

6

+

29647548

x

4

y

7

+

51883209

x

2

y

8

+

40353607

y

9