Exercise · ZW-BINOMIAL-COEFFICIENT-010 · ★★★ Advanced
How many subsets of \(\{1,2,\dots,8\}\) have an even number of elements (the empty set counts)?
Choose one option.
Show answer
- B. 128
Explanation
The sum \(\dbinom{8}{0}+\dbinom{8}{2}+\dots+\dbinom{8}{8}=1+28+70+28+1=128\), half of the \(2^8=256\) subsets.
Review the concept: Binomial coefficient
The binomial coefficient \(\binom{n}{k}\), for integers \(0\le k\le n\), is the number of \(k\)-element subsets of a set with \(n\) elements.
\[\binom{n}{k}=\frac{n!}{k!\,(n-k)!}\]
More practice problems: binomial coefficient
- ZW-BINOMIAL-COEFFICIENT-001 Compute dbinom62.
- ZW-BINOMIAL-COEFFICIENT-002 How many 3-element subsets does \1,2,3,4,5\ have?
- ZW-BINOMIAL-COEFFICIENT-003 Compute dbinom97.
- ZW-BINOMIAL-COEFFICIENT-004 Compute dbinom100+dbinom1010.
- ZW-BINOMIAL-COEFFICIENT-005 Compute dbinom73+dbinom74.
- ZW-BINOMIAL-COEFFICIENT-006 Compute dbinom2018.
- ZW-BINOMIAL-COEFFICIENT-007 Which expression equals dbinomnk for 0≤ k≤ n?
- ZW-BINOMIAL-COEFFICIENT-008 Compute ∑_k=0^5dbinom5k.