An ant starts at the point (0,0), and walks either one unit up, or one unit to the right, until it reaches the point (5,4). How many possible paths can the ant take?
An ant starts at the point (0,0), and walks either one unit up, or one unit to the right, until it reaches the point (5,4). How many possible paths can the ant take?
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
![**Problem 5:** An ant starts at the point (0, 0) and walks either one unit up, or one unit to the right, until it reaches the point (5, 4). How many possible paths can the ant take?
**Solution Explanation:**
To solve this problem, we can think of the ant's movement as a sequence of steps: the ant needs to make a total of 5 right moves and 4 up moves to reach the desired point. The order in which these moves are made determines the path.
The problem is equivalent to finding the number of ways to arrange a sequence of 5 "R" (right) moves and 4 "U" (up) moves. This is a combinatorial problem that can be solved using binomial coefficients.
The number of possible sequences can be calculated by finding the number of permutations of these 9 moves, which is given by:
\[ \text{Number of paths} = \binom{9}{5} \]
Here, \(\binom{9}{5}\) denotes the binomial coefficient, which represents the number of ways to choose 5 moves (R) out of a total of 9 moves (R + U). This is the same as choosing 4 moves (U) out of 9, \(\binom{9}{4}\), due to the symmetry of combinations.
The formula for the binomial coefficient is:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Applying this to the problem:
\[ \binom{9}{5} = \frac{9!}{5! \times 4!} = \frac{362880}{120 \times 24} = 126 \]
Therefore, there are 126 possible paths the ant can take from (0, 0) to (5, 4).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F19ccf01e-335f-4dab-a804-6828d515c4a4%2Fa05b2b87-a9a1-4542-a391-74bc25df75d3%2F1novdmj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5:** An ant starts at the point (0, 0) and walks either one unit up, or one unit to the right, until it reaches the point (5, 4). How many possible paths can the ant take?
**Solution Explanation:**
To solve this problem, we can think of the ant's movement as a sequence of steps: the ant needs to make a total of 5 right moves and 4 up moves to reach the desired point. The order in which these moves are made determines the path.
The problem is equivalent to finding the number of ways to arrange a sequence of 5 "R" (right) moves and 4 "U" (up) moves. This is a combinatorial problem that can be solved using binomial coefficients.
The number of possible sequences can be calculated by finding the number of permutations of these 9 moves, which is given by:
\[ \text{Number of paths} = \binom{9}{5} \]
Here, \(\binom{9}{5}\) denotes the binomial coefficient, which represents the number of ways to choose 5 moves (R) out of a total of 9 moves (R + U). This is the same as choosing 4 moves (U) out of 9, \(\binom{9}{4}\), due to the symmetry of combinations.
The formula for the binomial coefficient is:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Applying this to the problem:
\[ \binom{9}{5} = \frac{9!}{5! \times 4!} = \frac{362880}{120 \times 24} = 126 \]
Therefore, there are 126 possible paths the ant can take from (0, 0) to (5, 4).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 14 images
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,
