MATH SOLVE

4 months ago

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

Accepted Solution

A:

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]

[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]