Question

A jar contains 20 yellow jellybeans, 20 orange jellybeans, 20 red jellybeans and 20 green jellybeans.

Required:
a. In how many ways can you put all the jellybeans in a row?
b. How many ways are there to select a handful of 20 jellybeans?
c. How many ways are there to select a handful of 20 jellybeans that contains at least 3 red?
d. How many ways are there to select a handful of 20 jellybeans that contains at least 3 red and at most 2 orange?

Answer 1

A)

`\left \{ {{80} \atop {20}} \} * \left \{ {{60} \atop {20}} \} * \left \{ {{40} \atop {20}} \} * \left \{ {{20} \atop {20}} \}`

B)

`\left \{ {{20+4-1} \atop {4-1}} \} = \left \{ {{23} \atop {3}} \}`

C)

=
`\left \{ {{17+4+1} \atop {4-1}} \} = \left \{ {{20} \atop {3}} \}`

D)

=
`\left \{ {{20} \atop {3}} \} - \left \{ {{17} \atop {3}} \}`

Explanation:

A) How many ways can you put all Jellybeans in a row

Total number of Jellybeans = 80

The first jellybeans = 20 yellow , second is 20 orange jellybeans , third is 20 red jellybeans , fourth is 20 green jellybeans

Therefore the number of ways the Jellybeans can be put in a row is :

`\left \{ {{80} \atop {20}} \} * \left \{ {{60} \atop {20}} \} * \left \{ {{40} \atop {20}} \} * \left \{ {{20} \atop {20}} \}`

B) How many ways are there to select a handful of 20 jellybeans

lets assume:

yellow jellybeans = a , orange jellybeans = b , red jellybeans = c , green jellybeans = d

a + b + c + d = 20

This is the number Non-negative integer solutions

=
`\left \{ {{20+4-1} \atop {4-1}} \} = \left \{ {{23} \atop {3}} \}`

C) This is also the number of Non-negative integer solutions but in this case the value of C ≥ 3

hence the number of ways to select a handful of 20 jellybeans that contains at least 3 red

=
`\left \{ {{17+4+1} \atop {4-1}} \} = \left \{ {{20} \atop {3}} \}`

D) In this case the value of C ≥ 3 and B ≤ 2

Hence the number of ways to select a handful of 20 jellybeans that contains at least 3 red and at most 2 orange

=
`\left \{ {{20} \atop {3}} \} - \left \{ {{17} \atop {3}} \}`