Discussion
From a box of 20 balls, 8 balls are to be chosen. In how many ways can this be done ?
Answer: A
The number of ways is \({}^{20}C_{8}\).
Choosing 8 balls from 20 is a selection in which order does not matter: picking ball 1 and then ball 2 gives the same group as picking ball 2 and then ball 1. Selections are counted by combinations:
\[ {}^{20}C_{8} = \frac{20!}{8!\,12!} \]
\({}^{20}P_{8}\) counts ordered arrangements of 8 balls, which overcounts each group by 8! orderings. 20! is the number of ways to arrange all 20 balls in a row, and 8! the number of ways to arrange 8 balls, so neither counts selections of 8 from 20.
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