Answer :
Kim can arrange the 10 identical lamps on the 3 identical tables in 66 different ways.
Number of lamps= 10
Number of tables=3
For identical objects we have the following formula;
(n + r - 1)C(r - 1) (r - 1)
We thus have;
n + r - 1 = 10 + 3 - 1
= 12
r - 1 = 3 - 1
= 2
Thus,
(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!)
¹²C₂ = 12!/(10! * 2!)
¹²C₂ = 12 * 11 * 10!/(10! * 2!)
¹²C₂ = 132/2
= 66
Therefore, there are 66 ways to put all the lamps on the tables.
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