Find the Coefficient of x5y2 in (2x

Math Problem Statement

(a) Find the Coefficient of x5y2 in the Expansion of (2x-3y)7 (b) Find the Coefficient of x9y3 in the Expansion Of (x+2y)12 Determine the Coefficient of xyz2 in the Expansion of (2x-y-z)4 . a2b3c2d5 in the Expansion of (a-2b-3c=2d=5)16. (a) Explain about The Principle of Inclusion-Exclusion. (b) Consider a set of integers from 0 to 499 (i) Find the number of integers are divisible by 3 or 5 or 11. (ii)Find the number of integers are divisible by 3 0r 5 Not 11. (iii)Find the number of integers are divisible by 3 0r 11 Not 5. (iv)Find the number of integers cannot be divisible by 3 or 5 or 11. (a) Explain about Euler Graph. (b) Let “G” be a connected Planar Simple Graph with “e” edges “v” vertices, Let “r” be No .of Regions in Planar representation “G” Then Prove r=e-v=2. Explain about Working Rules of Isomorphism with Example. What is Minimum Spanning Tree? Explain about Kruskal’s Algorithm with Example

Solution

Let's tackle these problems one by one:

1. Binomial Expansion Coefficients

(a) Find the coefficient of $x^5y^2$ in the expansion of $(2x - 3y)^7$

To find the coefficient of $x^5y^2$ in the expansion of $(2x - 3y)^7$ , we use the binomial theorem:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

Here, $a = 2x$ , $b = -3y$ , and $n = 7$ . We need to find the term containing $x^5y^2$ .

$(2x)^5(-3y)^2 = 2^5 x^5 (-3)^2 y^2 = 32 x^5 \cdot 9 y^2 = 288 x^5 y^2$

The term in the expansion that contains $x^5 y^2$ corresponds to $k = 2$ :

$\binom{7}{2} (2x)^{5} (-3y)^{2} = \binom{7}{2} \cdot 2^5 \cdot (-3)^2 \cdot x^5 y^2 = \binom{7}{2} \cdot 32 \cdot 9 \cdot x^5 y^2$

[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \cdot 6}{2 \cdot 1} = 21 \

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Math Problem Analysis

Mathematical Concepts

Binomial theorem
Coefficient extraction

Formulas

Binomial theorem: (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Theorems

Suitable Grade Level

Advanced High School

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