34 dogs arrive at a kennel, but there are only 17 empty cages, in how many different ways can the dogs be placed?

To solve this problem, we need to find the number of ways to place 34 dogs into 17 cages.

This can be solved using the concept of combinations. Since the order of the dogs does not matter, and each dog can only be placed in one cage, we can use the formula for combinations.

The formula for combinations is given by:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

Where n is the total number of items (dogs in this case) and r is the number of items to be selected (cages in this case).

Here, we have n = 34 (number of dogs) and r = 17 (number of cages).

Substituting these values into the formula:

$$C(34, 17) = \frac{34!}{17!(34-17)!}$$

Simplifying:

$$C(34, 17) = \frac{34!}{17! \cdot 17!}$$

$$\frac{34!}{17! \cdot17!}=2333606220$$ .