Let's solve the problem step-by-step:
### Step 1: Understanding the Problem
We have five number cards: 21, 15, 14, 10, and 17. We need to find the probability that the sum of two randomly picked cards is more than 30.
### Step 2: Calculate the Total Number of Pairs
The number of ways to pick two cards out of five can be calculated using the combination formula [tex]\( \binom{n}{k} \)[/tex], where [tex]\( n \)[/tex] is the total number of items and [tex]\( k \)[/tex] is the number of items to choose.
For our problem:
[tex]\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\][/tex]
So, there are 10 possible pairs of cards.
### Step 3: List All Possible Pairs
The possible pairs of cards and their sums are:
1. (21, 15): sum is 36
2. (21, 14): sum is 35
3. (21, 10): sum is 31
4. (21, 17): sum is 38
5. (15, 14): sum is 29
6. (15, 10): sum is 25
7. (15, 17): sum is 32
8. (14, 10): sum is 24
9. (14, 17): sum is 31
10. (10, 17): sum is 27
### Step 4: Count Pairs with Sums Greater than 30
From the list above, identify pairs whose sums are greater than 30:
1. (21, 15): sum is 36
2. (21, 14): sum is 35
3. (21, 10): sum is 31
4. (21, 17): sum is 38
5. (15, 17): sum is 32
6. (14, 17): sum is 31
So, there are 6 pairs where the sum is greater than 30.
### Step 5: Calculate the Probability
The probability is the ratio of the favorable outcomes to the total outcomes. Here, the probability that the sum of two randomly picked cards is more than 30 is:
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{6}{10} = 0.6
\][/tex]
### Step 6: State the Result
The probability that the total of the two numbers is more than 30 is [tex]\(0.6\)[/tex] or 60%.
So, the final answer is:
- Total pairs: 10
- Pairs with sums greater than 30: 6
- Probability that the sum is more than 30: 0.6
