Question

the total number of distinct term in the expansion of (9x^2+4y^2-12xy)^8 is​

Answer 1

There are 45 distinct terms in the expansion of (9x² + 4y² - 12xy)⁸

Given

(9x² + 4y² - 12xy)⁸

To Find

Number of distinct terms

Solution

The individual terms that are separated by a plus '+' or a minus '-' sign in an expression are called distinct terms.

We know that for any Binomial expression (x₁ + x₂ + x₃ + ... + xₐ)ⁿ

The number of distinct terms in the binomial expansion will be ⁿ⁺ᵃ⁻¹Cₙ

Where,

n = exponential power

a = no. of terms in the bracket

Here, the Binomial expression is

(9x² + 4y² - 12xy)⁸

We can see that here

n = 8

a = 3

Therefore,

no. of distinct terms = ⁸⁺³⁻¹C₈

= ¹⁰C₈

= $\frac{10!}{2! X 8!}$                                                    [ⁿCₓ=n!/(x! (n - x)!)]

$=\frac{10 X 9 X 8!}{2 X 8!}$

= 5 X 9

= 45

Therefore, there are 45 distinct terms in the expansion of (9x² + 4y² - 12xy)⁸

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Answer 2

There are $45$different terms for the expansion of $(9x^{2} + 4y^{2} - 12xy)^{8}$

Given: In the given question we have given an expression which is:

$(9x^{2} + 4y^{2} - 12xy)^{8}$

Find:

Number of distinct expressions

Solution:

Individual expressions that are separated by a plus '+' or minus '-' sign in an expression are called distinct expressions.

We know that for any binomial expression $(x_{1} + x_{2} + x_{3} + ... + x_{a} )^{n}$

The total number of distinct terms in any binomial expansion can be calculated from $^{n+a-1} C_{n}$

where

n = exponential power

and = no. expressions in parentheses

In our question, we have given an expression that is $(9x^{2} + 4y^{2} - 12xy)^{8}$

Here is the binomial expression

$(9x^{2} + 4y^{2} - 12xy)^{8}$

We can see it here

$n = 8$

$a = 3$

Therefore, from the formula stated above

No. of different expressions =$^{8+3-1} C_{8}$

= $^{10} C_{8}$

$=\frac{10!}{2!*8!}\\\\= \frac{10*9*8!}{2*8!}$                             $(^nC_{x} =\frac{n!}{x!(n-x)!})$

$=5*9\\=45$

Therefore, there are 45 different terms in the expansion of $(9x^{2} + 4y^{2} - 12xy)^{8}$

Learn more here:

https://brainly.in/question/5121318

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