Question

Calculate the coefficients of the first four terms of the binomial expansion for the binomial (x + y)28

Answer 1

The coefficients of the first four terms of the expression are 1,28,378 and 3,276

Here, we want to get the coefficients of the first 4 terms

We start as follows;

`(x+y)^(28)=^(28)C_0x^(28)y^0+^(28)C_1x^(27)y^1+^(28)C_2x^(26)y^2+^(28)C_3x^(25)y^3\text{ + }\ldots..`

The coefficients are simply the combination parts

By calculating using the combinatorial formula, we can have the coefficients

The general formula for calculating the combination of two numbers is simply;

`^nC_r\text{ = }(n!)/((n-r)!r!)`

We now proceed to apply the formula above to each of the combination expressions in the expansion

We have this as follows;

`\begin{gathered} ^(28)C_0\text{ = }(28!)/((28-0)!0!)\text{ = }1 \\ \\ ^(28)C_1\text{ = }(28!)/((28-1)!1!)\text{ = 28} \\ \\ ^(28)C_2\text{ = }(28!)/((28-2)!2!)\text{ = 378} \\ \\ ^(28)C_3\text{ = }(28!)/((28-3)!3!)\text{ = 3,276} \end{gathered}`