Exercise · ZW-BINOMIAL-THEOREM-006 · ★★☆ Intermediate
In the expansion of \((x+y)^6\), what is the coefficient of the middle term \(x^3y^3\)?
Choose one option.
Show answer
- B. 20
Explanation
The coefficient is \(\binom{6}{3}=\frac{6!}{3!\,3!}=20\). The 15 is \(\binom{6}{2}\), a neighboring term, and 18 is \(3\cdot 6\).
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
- ZW-BINOMIAL-THEOREM-001 Find the coefficient of x^2 in the expansion of (x+1)^5.
- ZW-BINOMIAL-THEOREM-002 How many terms does the expansion of (a+b)^8 have?
- 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?
- 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.