To find the number of different sets of 4 marbles that can be picked from the bag, we use the concept of combinations. The formula to calculate combinations is given by:

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

where:

• $n$ is the total number of items to choose from.
• $r$ is the number of items to choose.

First, let's calculate the total number of marbles in the bag:

$n = 3 \ (\text{red}) + 2 \ (\text{green}) + 1 \ (\text{purple}) + 2 \ (\text{yellow}) + 2 \ (\text{orange}) = 10$

So, the total number of marbles $n = 10$.

Now, we need to find the number of ways to pick 4 marbles at a time:

$C(10, 4) = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$

Thus, the number of different sets of picking up 4 marbles at a time is 210.

Would you like details on the calculation or have any questions?

Here are 5 related questions:

1. How would the result change if the number of marbles of each color were different?
2. What if you needed to pick 5 marbles instead of 4?
3. How would you calculate the probability of picking exactly one marble of each color?
4. What if the marbles were picked with replacement? How would that change the outcome?
5. How does the formula for combinations differ from that of permutations?

Tip: When calculating combinations, remember that the order of selection does not matter. If the order matters, you would use permutations instead.