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;

$^nC_r\text{ = }\frac{n!}{(n-r)!r!}$

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

We have this as follows;

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