Question

Rachel is arranging 12 cans of food in a row on a shelf. She has 8 cans of beets, 3 cans of corn, and 1 can of beans. In how many distinct orders can the cans be arranged if two cans of the same food are considered identical (not distinct)?

Answer 1

Given:

Total number of food cans = 12

Cans of Beets = 8

Cans of corn = 3

Can of beans = 1

To find:

How many distinct orders can the cans be arranged if two cans of the same food are considered identical.

Solution:

To find the distinct ways arrangement, we have a formula:

`\text{Number of distinct ways}=(n!)/(r_1!r_2!...r_k!)` ...(i)

Where, n is the number of objects and
`r_1,r_2,...,r_k` are repeated objects.

Total number of food cans is 12. So,
`n=12`.

She has 8 cans of beets. So,
`r_1=8`

She has 3 cans of corns. So,
`r_2=3`

She has 1 can of beans. So,
`r_3=1`

Substituting these values in (i), we get

`\text{Number of distinct ways}=(12!)/(8!3!1!)`

`\text{Number of distinct ways}=(12* 11* 10* 9* 8!)/(8!* 3* 2* 1* 1)`

`\text{Number of distinct ways}=(11880)/(6)`

`\text{Number of distinct ways}=1980`

Therefore, the number of distinct orders to arrange the cans is 1980.