Question

Kim has $10$ identical lamps and $3$ identical tables. How many ways are there for her to put all the lamps on the tables

Answer 1

There are 66 ways Kim can put the 10 identical lamps on the 3 identical tables

How to determine the number of ways?

The given parameters are:

• Identical lamps, n = 10
• Identical tables, r = 3

The combination involving identical objects is calculated using:

(n + r - 1)C(r - 1)

So, we have:

n + r - 1 = 10 + 3 - 1

Evaluate the sum

n + r - 1 = 13 - 1

Evaluate the difference

n + r - 1 = 12

Also, we have:

r - 1 = 3 - 1

Evaluate the difference

r - 1 = 2

So, we have:

(n + r - 1)C(r - 1) = 12C2

Apply the following combination formula:

nCr = n!/((n - r)!r!)

So, we have:

12C2 = 12!/((12 - 2)! * 2!)

Evaluate the difference

12C2 = 12!/(10! * 2!)

Expand the numerator

12C2 = 12 * 11 * 10!/(10! * 2!)

Evaluate the quotient

12C2 = 12 * 11/2!

Expand the denominator

12C2 = 12 * 11/2 * 1

Evaluate the product

12C2 = 132/2

Evaluate the quotient

12C2 = 66

Hence, there are 66 ways Kim can put the 10 identical lamps on the 3 identical tables

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