Question

pritika is using the binomiak theorem to find tbe binomial expansion of (a+b)^7 which of the following is a term in the expansion

Answer 1

To find a term in the binomial expansion of \((a + b)^7\) using the Binomial Theorem, we use the formula:

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

where \(\binom{n}{k}\) is the binomial coefficient, defined as \(\frac{n!}{k!(n-k)!}\).

For \((a + b)^7\), we have \(n = 7\). The general term in the expansion is given by:

\[
T_k = \binom{7}{k} a^{7-k} b^k
\]

where \(k\) can take values from 0 to 7.

To determine a specific term, you can substitute a particular value of \(k\). For example:

- If \(k = 0\):
\[
T_0 = \binom{7}{0} a^{7} b^{0} = 1 \cdot a^7 = a^7
\]

- If \(k = 1\):
\[
T_1 = \binom{7}{1} a^{6} b^{1} = 7a^6b
\]

- If \(k = 2\):
\[
T_2 = \binom{7}{2} a^{5} b^{2} = 21a^5b^2
\]

- If \(k = 3\):
\[
T_3 = \binom{7}{3} a^{4} b^{3} = 35a^4b^3
\]

You can calculate further for \(k = 4, 5, 6, 7\) if needed. Each of these expressions represents a term in the binomial expansion of \((a + b)^7\).

If you would like to identify a specific term or coefficient from the expansion, please let me know which \(k\) value or a specific term you're interested in!