Exercise · ZW-BINOMIAL-COEFFICIENT-008 · ★★☆ Intermediate
Compute \(\displaystyle\sum_{k=0}^{5}\dbinom{5}{k}\).
Choose one option.
Show answer
- A. 32
Explanation
The sum counts all subsets of a 5-element set, whatever their size: \(2^5=32\). The value \(31\) forgets the empty subset.
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-009 A grid is traversed from the bottom-left corner to the top-right corner using 4 steps right and 3 s…