Question
upstudy study bank question image url

A baseball team's roster consists of 25 players with 13 position players and the rest being pitchers. The team's batting order for a particular game consists of 9 players. The first 8 places in the batting order are taken by position players, The last place is taken by a pitcher. How many batting orders are possible? Use the formula for permutations to find your answer. 15,444 2,042,975 \( 51,891,840 \) \( 622,702,080 \)

Ask by Edwards Collins. in the United States
Jan 22,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

There are 622,702,080 possible batting orders.

Solution

To determine the number of possible batting orders under the given constraints, we'll break down the problem step by step. ### **Given:** - **Total players:** 25 - **Position players:** 13 - **Pitchers:** 25 - 13 = 12 ### **Batting Order Requirements:** - **First 8 spots:** Must be occupied by position players. - **9th spot:** Must be occupied by a pitcher. ### **Calculations:** 1. **Choosing and Arranging the First 8 Position Players:** - We need to select 8 out of the 13 position players and arrange them in the first 8 spots. - This can be calculated using permutations: \( P(13, 8) = 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \) - \( P(13, 8) = 51,891,840 \) 2. **Selecting the Pitcher for the 9th Spot:** - We have 12 pitchers to choose from for the last spot. - So, there are 12 possible choices. 3. **Total Number of Batting Orders:** - Multiply the number of ways to arrange the position players by the number of choices for the pitcher. - Total batting orders = \( P(13, 8) \times 12 = 51,891,840 \times 12 = 622,702,080 \) ### **Final Answer:** \(\boxed{622,\!702,\!080}\)

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Beyond the Answer

To calculate the number of possible batting orders, we need to arrange the first 8 position players and then choose 1 pitcher for the last spot. First, we can choose 8 position players from 13. The number of ways to choose 8 players is given by the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \): \[ C(13, 8) = \frac{13!}{8!5!} = 1,287. \] Next, we arrange the 8 chosen position players, which can be done in \( 8! = 40,320 \) ways. For the pitcher, we have 12 options (since there are 12 pitchers left). So, the total number of batting orders is: \[ C(13, 8) \times 8! \times 12 = 1,287 \times 40,320 \times 12 = 622,702,080. \] Hence, the answer is \( 622,702,080 \).

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy