There are twenty numbered balls in a bag.Two of the balls are numbered 0,six are numbered 1, five are numbered 2 and seven are numbered 3.
Four of these balls are chosen at random,without replacement.Calculate the different number of ways this can be done so that the four balls have that total 3.
70
To calculate the different number of ways to choose four balls from the bag such that their total is 3, we need to consider all possible combinations of balls with different numbers.
Let's break it down step-by-step:
Step 1: Calculate the number of combinations with one ball numbered 0 and three balls numbered 3.
- There are 2 balls numbered 0.
- There are 7 balls numbered 3.
To choose one ball numbered 0, we have 2 options.
To choose three balls numbered 3, we have 7C3 = 35 options.
Total combinations for this step: 2 * 35 = 70.
Step 2: Calculate the number of combinations with two balls numbered 1 and two balls numbered 2.
- There are 6 balls numbered 1.
- There are 5 balls numbered 2.
To choose two balls numbered 1, we have 6C2 = 15 options.
To choose two balls numbered 2, we have 5C2 = 10 options.
Total combinations for this step: 15 * 10 = 150.
Step 3: Calculate the total number of combinations.
Total combinations = combinations in Step 1 + combinations in Step 2 = 70 + 150 = 220.
Therefore, there are 220 different ways to choose four balls from the bag such that their total is 3.
To calculate the different number of ways to choose four balls such that the total is 3, we need to consider the different combinations of numbered balls that can add up to 3.
Let's break down the possibilities step by step:
1. Choosing one ball numbered 0:
- Since there are two balls numbered 0 in the bag, we can choose any one of them.
- Next, we need to choose three more balls with a total of 3. The sum of three balls cannot be 3 since the minimum sum of three balls in this bag would be 0 + 1 + 1 = 2.
- Therefore, there are no possible combinations in this case.
2. Choosing two balls numbered 0:
- There are two balls numbered 0, so we can choose any two of them.
- Now, we need to choose the remaining two balls that add up to 3 - 0 - 0 = 3.
- Since there are seven balls numbered 3, we have to consider different cases:
- Case 1: Choosing two balls numbered 3. There are 7 choose 2 ways to select two balls from the seven available.
- Case 2: Choosing one ball numbered 3 and one ball numbered 2. There are 7 ways to select a ball numbered 3 and 5 ways to select a ball numbered 2. So, the total number of combinations in this case is 7 * 5.
- Case 3: Choosing one ball numbered 3 and one ball numbered 1. Similarly, there are 7 ways to select a ball numbered 3 and 6 ways to select a ball numbered 1. So, the total number of combinations in this case is 7 * 6.
- Adding up the combinations from each case: 7 choose 2 + 7 * 5 + 7 * 6.
3. Choosing three balls numbered 0:
- There are two balls numbered 0, so we can choose any three of them.
- Now, we need to choose the remaining one ball that adds up to 3 - 0 - 0 - 0 = 3.
- Since there are five balls numbered 2, we have 5 ways to select a ball numbered 2.
4. Choosing four balls numbered 0:
- Since there are only two balls numbered 0, we can't choose four balls numbered 0.
Adding up the results from all cases, the different number of ways to choose four balls with a total of 3 is given by:
7 choose 2 + 7 * 5 + 7 * 6 + 5 = 21 + 35 + 42 + 5 = 103.
Therefore, there are 103 different ways to choose four balls such that the total is 3.