A bag has 4 red, 6 white, and 9 blue marbles. How many ways can 5 marbles be selected if 2 are one color, and 3 are another color?
could be
RRWWW
RRBBB
WWRRR
WWBBB
BBRRR
BBWWW
To calculate the number of ways to select 5 marbles with 2 of one color and 3 of another color, we can break it down into two steps:
Step 1: Select the color combination
Step 2: Count the number of ways to select marbles of each color
Step 1: Select the color combination
Since we need to select 2 marbles of one color and 3 marbles of another color, we can select the colors in two ways - either 2 red and 3 white marbles or 2 white and 3 red marbles.
Step 2: Count the number of ways to select marbles of each color
For the first color combination, let's consider selecting 2 red and 3 white marbles.
There are 4 red marbles to choose from and we need to select 2 of them. This can be done in the following number of ways:
4C2 = (4!)/(2!(4-2)!) = (4!)/(2!2!) = (4x3)/(2x1) = 6
Similarly, there are 6 white marbles to choose from, and we need to select 3 of them. This can be done in the following number of ways:
6C3 = (6!)/(3!(6-3)!) = (6!)/(3!3!) = (6x5x4)/(3x2x1) = 20
Therefore, for the first color combination, there are 6 ways to select 2 red marbles and 20 ways to select 3 white marbles.
Similarly, for the second color combination (2 white and 3 red marbles), there will be the same number of ways to select marbles for each color, i.e., 6 ways to select 2 white marbles and 20 ways to select 3 red marbles.
Hence, the total number of ways to select 5 marbles with 2 of one color and 3 of another color will be the sum of these two combinations:
6 + 20 = 26 ways.
Therefore, there are 26 ways to select 5 marbles with 2 of one color and 3 of another color from the given bag.