Question

1. Assume there are 5 red balls, 6 blue balls, and 4 green balls. If the balls are removed from the box one at a time, in how many different orders can the balls be removed assuming two balls of the same type are indistinguishable.

2. Give a recursive definition of the set of all even positive integers not divisible by 3.

Please write clearly so I can study from it! Try not to skip steps as much as you can.

Thank you!

Answer 1

Sol:

(1)

First, note that there are 15! permutations of the balls when the 5 red balls, 6 blue balls, and 4 green balls are distinguished from each other.

Then, the number of permutations of these 15 balls is 15! / (5!6!4!) = 630630

Therefore, there are 630630 different orders can the balls be removed.

(2) The set of all even positive integers not divisible by 3.

Basis Step:

Let x ∈ Z⁺ and x is not divisible by 3.

⇒ x ≡ k (mod 3), k = 1, 2.

So, the initial set of elements can be given as 1 ∈ S and 2 ∈ S.

Recursive Step:

To generate the new elements from that known to be in the set is:

If x ∈ S, then x + 3 ∈ S.

Therefore, there is no element in S unless it is obtained from the Basis and Recursive steps.