Which of these is not a variable term found in the binomial expansion of $(a+b)^8$? (Ignore coefficients.)
The binomial theorem states that the expansion of \((a+b)^n\) consists of terms of the form \(C(n, k) a^{n-k} b^k\), where \(n\) is a non-negative integer and \(k\) is an integer such that \(0 \le k \le n\). In this expansion, the sum of the exponents of \(a\) and \(b\) in every term must be equal to the power \(n\). For the expansion of \((a+b)^8\), the power is \(n=8\). Therefore, every term in the expansion must have the form \(a^i b^j\) where \(i + j = 8\) and \(i, j \ge 0\). We need to examine the given options to see which one does not satisfy this condition.
We check the sum of exponents for each of the given options:
Comparing these sums to the requirement that the sum must be 8, we see that options b, c, and d satisfy the condition, while option a does not.
Since the sum of the exponents in option (a) is 7, it does not match the required power of 8 for the binomial expansion of \((a+b)^8\). Therefore, \(a b^6\) is not a term in the expansion. The final answer is option (a).
Understanding binomial expansions is essential in fields like probability theory and statistics. For instance, when you toss a coin multiple times, the binomial theorem helps in determining the likelihood of getting a specific number of heads and tails. Recognizing which terms belong to an expansion allows you to quickly identify possible outcomes in complex probabilistic models or simplified physical simulations involving systems with two primary states.