A gallery has 25 paintings in its permanent collection, with display space for 10 at one time. How many different collections can be shown?

You need to know in how many ways you can choose 10 out of 25 items when order does not matter. This is a combination problem.

[tex] _{n}C_{r} = \dfrac{n!}{(n - r)!r!} [/tex]

[tex] _{25}C_{10} = \dfrac{25!}{(25 - 10)!10!} [/tex]

[tex] _{25}C_{10} = \dfrac{25 \times 24 \times 23 \times 22 \times 21 \times 20 \times 19 \times 18 \times 17 \times 16 \times 15!}{15! \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} [/tex]

[tex] _{25}C_{10} = \dfrac{25 \times 24 \times 23 \times 22 \times 21 \times 20 \times 19 \times 18 \times 17 \times 16}{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} [/tex]

[tex] _{25}C_{10} = \dfrac{23 \times 11 \times 20 \times 19 \times 2 \times 17}{1} [/tex]

[tex] _{25}C_{10} = 23 \times 11 \times 20 \times 19 \times 2 \times 17 [/tex]

[tex] _{25}C_{10} = 3,268,760 [/tex]