Math Problem Statement

The bag contains 3 red, 2 green, 1 purple, 2 yellow, and 2 orange marbles.Find the number of sets of picking up 4 marbles at a time that contain all the red marbles available.

Solution

To find the number of sets of picking up 4 marbles that contain all the 3 red marbles, we need to first consider that 3 of the 4 marbles are already determined (they must be red). This leaves us with the task of choosing 1 additional marble from the remaining non-red marbles.

Here's the breakdown:

  1. Number of red marbles: 3 (which must be included in the set).
  2. Total number of non-red marbles: 103=710 - 3 = 7.

These 7 non-red marbles consist of:

  • 2 green
  • 1 purple
  • 2 yellow
  • 2 orange

Now, we need to choose 1 marble from these 7 non-red marbles.

The number of ways to choose 1 marble from 7 is calculated as follows:

C(7,1)=7!1!(71)!=7C(7, 1) = \frac{7!}{1!(7-1)!} = 7

Thus, the number of sets of picking up 4 marbles that contain all 3 red marbles is 7.

Would you like more details on this calculation or have any other questions?

Here are 5 related questions:

  1. How would the result change if there were only 2 red marbles?
  2. What if you were asked to pick 5 marbles containing all the red ones?
  3. How many sets of 4 marbles can be picked if the set must include exactly 2 red marbles?
  4. What is the probability of picking a set of 4 marbles that includes all the red ones?
  5. How does the number of sets change if more colors or marbles were added?

Tip: When dealing with combinations, it helps to separate the problem into smaller steps—first fixing the required items, then considering the choices left for the remaining items.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Combinations

Formulas

Combination formula: C(n, k) = n! / (k! * (n - k)!)

Theorems

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Suitable Grade Level

High School