TAOCP 1.2.6 Exercise 1

By the symmetry condition (6), \binom{n}{n-1}=\binom{n}{1}.

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Section 1.2.6: Binomial Coefficients

Exercise 1. [00] How many combinations of $n$ things taken $n-1$ at a time are possible?

Verified: yes
Solve time: 38s

By the symmetry condition (6),

$$ \binom{n}{n-1}=\binom{n}{1}. $$

By (4),

$$ \binom{n}{1}=n. $$

Therefore the number of combinations of $n$ things taken $n-1$ at a time is

$$ \boxed{n}. $$

Indeed, each such combination is obtained by omitting exactly one of the $n$ objects.