Question

How many ways can I select 6 black and 3 blue marbles that is 17 black and 9 blue?

Answer 1

There are 1,039,584 ways in which this can be done.

Explanation:

Total number of black marbles = 17

Total number of blue marbles = 9

The number black marbles to be chosen = 6

The number blue marbles to be chosen = 3

So, we have to choose 6 black from 17 black marbles.

and to choose 3 blue from 9 blue marbles.

So, the number if possible ways to do it :
`^(17)  \textrm{C}_ {6}  * ^(9)  \textrm{C}_ {3}`

`^(n)  \textrm{C}_ {r}  = (n! )/(r! (n-r)!)`

Now, solving
`^(17)  \textrm{C}_ {6}`, we get:

`^(17)  \textrm{C}_ {6}  = (17!)/(6! * 11!)  \\= (17 * 16 * 15 * 14* 13* 12 * 11! )/(11! * (6* 5 * 4 * 3* 2))   = 12,376`

solving
`^(9)  \textrm{C}_ {3}`, we get:

`^(9)\textrm{C}_(3)  = (9!)/(6! * 3!)\\= (9 * 8 * 7  * 6! )/(6!* (3* 2 )) = 84`

`\implies^(17)\textrm{C}_ {6}* ^(9)  \textrm{C}_ {3}=12,376 * 84 = 1,039,584`

Hence, there are 1,039,584 ways in which this can be done.