Math Problem Statement

answer this with illustration

Answer the following three math problems with illustrations: 1. If 3 people are to be arranged in a sofa for a photo shoot, how many arrangements are possible? 2. Your boss asked you to prepare the venue for a concert with audience seating in a triangular manner. The first row has 2 chairs, increasing by 2 chairs per row for 26 rows. How many chairs are needed? 3. There are 3 routes from City A to City B and 2 routes from City B to City C. How many total routes are possible?

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:

P(n,r)=n!P(n, r) = n!

For n=3n = 3:

P(3,3)=3!=3×2×1=6P(3, 3) = 3! = 3 \times 2 \times 1 = 6

Illustration:

  1. 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: an=a1+(n1)da_n = a_1 + (n-1)d Where:

  • a1=2a_1 = 2 (first row has 2 chairs),
  • d=2d = 2 (difference per row).

The total number of chairs is: Sn=n2×(a1+an)S_n = \frac{n}{2} \times (a_1 + a_n) For n=26n = 26: a26=2+(261)×2=52a_{26} = 2 + (26 - 1) \times 2 = 52 S26=262×(2+52)=13×54=702S_{26} = \frac{26}{2} \times (2 + 52) = 13 \times 54 = 702

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: Total Routes=Routes from A to B×Routes from B to C\text{Total Routes} = \text{Routes from A to B} \times \text{Routes from B to C} 3×2=63 \times 2 = 6

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:

  1. How would the arrangement change if 4 people are to be seated instead of 3?
  2. How does the total number of chairs change if the rows increase to 30?
  3. What happens if the difference dd in the triangular seating arrangement is 3 instead of 2?
  4. Can we generalize the formula for the number of routes in Problem 3 for any number of routes between cities?
  5. 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