Henry Ernest Dudeney/Modern Puzzles/166 - Picture Presentation/Solution/Proof 2

by : $166$

 * Picture Presentation

Proof
The number of ways you can pick $r$ pictures from a collection of $10$ is equal to the binomial coefficient $\dbinom n r$, which is given by:
 * $\dbinom n r = \dfrac {n!} {r! \paren {n - r}!}$

Hence the total number of ways the collector can make part or all of his collection available to the public is:
 * $\ds \sum_{k \mathop = 1}^n \dbinom {10} k$

which is equal to:


 * $\ds \sum_{k \mathop = 0}^n \dbinom {10} k - \dbinom {10} 0$

From Sum of Binomial Coefficients over Lower Index:
 * $\ds \sum_{k \mathop = 0}^n \dbinom {10} k = 2^{10} = 1024$

while from Binomial Coefficient with Zero:
 * $\dbinom {10} 0$

Hence if the collector wants to consider the option of not giving any of his paintings after all, the number of options he has is $2^{10} = 1024$.

If he is committed to giving at least $1$, then the option of giving none is off the table, so the answer is $1023$.