Solution: 2013 Fall Midterm - 14
Author: Pat MorinQuestion
What does the summation $ \sum_{k=7}^{n} \binom{k-1}{6} $ count?
(a)
The numbre of subsets of ${1, 2, 3, ..., n}$ having size 5
(b)
The numbre of subsets of ${1, 2, 3, ..., n}$ having size 7
(c)
The numbre of subsets of ${1, 2, 3, ..., n}$ having size 6
(d)
None of the above.
Solution
For answer c), if we sub in $n = 7$, then we are trying to find the number of subsets of ${1, 2, … 7 }$ having size 7.
There is only 1 way to do this (choosing all the numbers in the set ${1, 2, … 7 }$, which matches the summation $\sum_{k=7}^{n} \binom{k-1}{6} = \sum_{k=7}^{7} \binom{k-1}{6} = 1$.
For the options a) and b), there are more than one subset having size 5 and size 6, which means it doesn’t match the summation in the question description.