Final answer:
To determine the number of ways to select four balls with at least two volleyballs from the sack, we need to calculate combinations for the different case scenarios—selecting two, three, or four volleyballs—and then sum the results to find the total number of ways.
Step-by-step explanation:
The question asks us to find the number of ways to select four balls from a sack containing six basketballs and five volleyballs, with the condition that at least two of the balls selected are volleyballs. This is a combinatorial mathematics problem which can be solved using combinations. To solve this, we need to consider the different scenarios where the condition of selecting at least two volleyballs is satisfied:
1. Selecting exactly two volleyballs and two basketballs
2. Selecting exactly three volleyballs and one basketball
3. Selecting all four volleyballs
For each scenario, we calculate the combinations separately and then sum them up to find the total number of ways.
Case 1: Two volleyballs and two basketballs: This can be done in combination of five volleyballs taken two at a time times combination of six basketballs taken two at a time.
Case 2: Three volleyballs and one basketball: This can be done in combination of five volleyballs taken three at a time times combination of six basketballs taken one at a time.
Case 3: All four volleyballs: There is only one way to select all four volleyballs, as there are only five volleyballs in total.
Adding the results from all three cases will give us the total number of ways to select the balls under the given condition.