Exercise · ZW-BINOMIAL-COEFFICIENT-007 · ★★☆ Intermediate
Which expression equals \(\dbinom{n}{k}\) for \(0\le k\le n\)?
Choose one option.
Show answer
- D. \(\dbinom{n}{n-k}\)
Explanation
Choosing \(k\) elements is the same as choosing the \(n-k\) left out: \(\dbinom{n}{k}=\dbinom{n}{n-k}\). The others are not equal in general (with \(n=4,\,k=1\) the value is \(4\) and they give \(3\), \(6\) and \(24\)).
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-008 Compute ∑_k=0^5dbinom5k.
- 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…