What is the coefficient of x2 in (1 + x)11? Describe how relation can be represented using matrix.

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Solution:

In Short:

11C2 = \(\frac{11!}{2! \times (11 – 2)!}\) = 55

Explanation:

In the example of (a + b)n, the (r + 1)th term, denoted by tr + 1, is given by

tr + 1nCr . an-r . br

Letting a = 1, b = x and n = 11, we get

tr + 1 = 11Cr . 111-r . xr

As, we need the cofficient of x2, we have to take r = 2

11C2 = \(\frac{11!}{2! \times (11 – 2)!}\) = 55

The coefficient of x2 in (1 + x)11 is 55.

Representaton of Relation using Matrix:

Let A = {a1, a2, …….., am} and B = {b1, b2, ………, bn) are finite sets containing m and n elements respectively and set R be a relation from A to B then R can be represented by the mn matrix.

MR = [mij] where mij = \(\left\{\begin{matrix}1 \enspace if \enspace (a_i, b_j) \enspace ∈ \enspace R \\ 0 \enspace if \enspace (a_i, b_j) \enspace ∉ \enspace R\end{matrix}\right.\)

The matrix MR is called matrix of R.

Example:

Let A = {1, 3, 4}, R = {(1, 1), (1, 3), (3, 3), (4, 4)}

MR = \(\begin{bmatrix}1 & 1 & 0\\ 0 & 1& 0\\ 0 & 0 & 1\end{bmatrix}\)

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