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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)?

User Jojonas
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1 Answer

3 votes

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.

User Ram Narasimhan
by
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