M/C How many solutions are there to the equation X1+ X2 + X3 + X4 + Xg = 25 where each X¡ is and integer that satisfies 0sx;s 1O? p ( 29 18 4 4 p( 29 18 -5 4 4 4 p( 29 18 +5. 4 -5· 4 4
M/C How many solutions are there to the equation X1+ X2 + X3 + X4 + Xg = 25 where each X¡ is and integer that satisfies 0sx;s 1O? p ( 29 18 4 4 p( 29 18 -5 4 4 4 p( 29 18 +5. 4 -5· 4 4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
please seea ttached
![**Problem Statement:**
**M/C**
How many solutions are there to the equation
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 25 \]
where each \( x_i \) is an integer that satisfies \( 0 \leq x_i \leq 10 \)?
**Answer Choices:**
1. \( \binom{29}{4} - \binom{18}{4} \)
2. \( \binom{29}{4} - 5 \cdot \binom{18}{4} + \left(\binom{5}{2}\right) \cdot \binom{7}{4} \)
3. \( \binom{29}{4} - 5 \cdot \binom{18}{4} + 5 \cdot \binom{7}{4} \)
4. \( \binom{29}{4} - 5 \cdot \binom{18}{4} \)
**Explanation:**
The problem asks to find the number of non-negative integer solutions to a linear equation under given constraints. The constraints specify that each variable can take values from 0 to 10. The solution involves combinatorial methods, often involving binomial coefficients to count the number of valid combinations. Each choice represents a different way of calculating these combinations, typically involving inclusion-exclusion principles or transformations to ensure constraints are met.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64890eb5-4eb0-4853-894d-dd9980127f88%2Fcc9b6e12-c4b1-4ec9-9a9a-53156bf47786%2Frcnexpg_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
**M/C**
How many solutions are there to the equation
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 25 \]
where each \( x_i \) is an integer that satisfies \( 0 \leq x_i \leq 10 \)?
**Answer Choices:**
1. \( \binom{29}{4} - \binom{18}{4} \)
2. \( \binom{29}{4} - 5 \cdot \binom{18}{4} + \left(\binom{5}{2}\right) \cdot \binom{7}{4} \)
3. \( \binom{29}{4} - 5 \cdot \binom{18}{4} + 5 \cdot \binom{7}{4} \)
4. \( \binom{29}{4} - 5 \cdot \binom{18}{4} \)
**Explanation:**
The problem asks to find the number of non-negative integer solutions to a linear equation under given constraints. The constraints specify that each variable can take values from 0 to 10. The solution involves combinatorial methods, often involving binomial coefficients to count the number of valid combinations. Each choice represents a different way of calculating these combinations, typically involving inclusion-exclusion principles or transformations to ensure constraints are met.
Expert Solution
Step 1
The number of solutions can be calculated using the star and sticks method. The formula for this method is .
Here, n is the solution or the right hand side value of the equation and r is the total number of variables. Use the formula to obtain the number of solutions.
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
