Exercise · ZW-BINOMIAL-THEOREM-001 · ★☆☆ Basic
Find the coefficient of \(x^2\) in the expansion of \((x+1)^5\).
Choose one option.
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- C. 10
Explanation
The term with \(x^2\) is \(\binom{5}{2}x^2\cdot 1^3\), and \(\binom{5}{2}=10\). The 5 would be the coefficient of \(x\) and 20 comes from forgetting to divide by \(2!\).
Review the concept: Binomial theorem
The binomial theorem expresses the power \((x+y)^n\), with \(n\) a nonnegative integer, as a sum of terms whose coefficients are the binomial coefficients \(\binom{n}{k}\).
\[(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}\]
More practice problems: binomial theorem
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- ZW-BINOMIAL-THEOREM-003 Find the coefficient of x^3 in (x+2)^5.
- ZW-BINOMIAL-THEOREM-004 Find the coefficient of x^2 in (2x-1)^4.
- ZW-BINOMIAL-THEOREM-005 What is ∑_k=0^7binom7k?
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- ZW-BINOMIAL-THEOREM-007 Expand (x-y)^3.
- ZW-BINOMIAL-THEOREM-008 Find the constant term of the expansion of (x+1x)^6.
- ZW-BINOMIAL-THEOREM-009 Compute ∑_k=0^8(-1)^kbinom8k.