Math Problem Statement
answer this with illustration
Solution
PIL import Image from IPython.display import display
Open the image to analyze the text and create illustrations for the solutions
image_path = "/mnt/data/1733061437295x3nv3ujc.jpg" img = Image.open(image_path) display(img)Let's solve each problem step by step with illustrations:
Problem 1:
"If 3 people are to be arranged in a sofa for a photo shoot, how many arrangements are possible?"
The number of arrangements is given by the formula for permutations:
For :
Illustration:
- Person A, B, C can sit as:
- ABC
- ACB
- BAC
- BCA
- CAB
- CBA
Problem 2:
"Audience sitting in a triangular manner, starting with 2 chairs in the first row and increasing by 2 chairs per row, up to 26 rows. How many chairs are needed?"
The number of chairs forms an arithmetic sequence: Where:
- (first row has 2 chairs),
- (difference per row).
The total number of chairs is: For :
Illustration:
- Row 1: 2 chairs
- Row 2: 4 chairs
- Row 3: 6 chairs
- ...
- Row 26: 52 chairs.
Problem 3:
"Routes from City A to City C via City B: 3 routes from A to B, 2 routes from B to C. How many total routes?"
Multiply the number of routes:
Illustration:
- Route combinations:
- A1-B1-C1, A1-B1-C2,
- A2-B2-C1, A2-B2-C2,
- A3-B1-C1, A3-B1-C2.
Let me know if you'd like detailed diagrams or further clarifications! Here are five additional related questions:
- How would the arrangement change if 4 people are to be seated instead of 3?
- How does the total number of chairs change if the rows increase to 30?
- What happens if the difference in the triangular seating arrangement is 3 instead of 2?
- Can we generalize the formula for the number of routes in Problem 3 for any number of routes between cities?
- How can combinations be used if the order of seating in Problem 1 doesn't matter?
Tip: Always double-check patterns and formulas in such problems for clarity!
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Math Problem Analysis
Mathematical Concepts
Permutations
Arithmetic Sequence
Counting Principles
Formulas
P(n, r) = n! (permutation formula)
a_n = a_1 + (n-1)d (nth term of arithmetic sequence)
S_n = (n/2) * (a_1 + a_n) (sum of arithmetic sequence)
Total Routes = Routes from A to B * Routes from B to C
Theorems
Fundamental Counting Principle
Properties of Arithmetic Sequences
Suitable Grade Level
Grades 6-9
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