Pascal's Triangle Problems and Solutions

Generate rows of Pascal's Triangle, essential for binomial expansion and combinatorics. Select the number of rows to explore Pascal's Triangle for pattern recognition and binomial applications.

Problem 1: Find the value at row 4, column 2 in Pascal's Triangle.

Solution:

The value at row 4, column 2 is found using the binomial coefficient formula: C(n, k) = n! / (k!(n - k)!)

For n = 4 and k = 2:

C(4, 2) = 4! / (2!(4 - 2)!) = 24 / (2 × 2) = 6

Answer: 6

Problem 2: Find the value at row 6, column 3 in Pascal's Triangle.

Solution:

C(6, 3) = 6! / (3!(6 - 3)!) = 720 / (6 × 6) = 20

Answer: 20

Problem 3: Find the value at row 7, column 0 in Pascal's Triangle.

Solution:

The first element in each row of Pascal's Triangle is always 1. So, C(7, 0) = 1.

Answer: 1

Problem 4: Find the value at row 8, column 4 in Pascal's Triangle.

Solution:

C(8, 4) = 8! / (4!(8 - 4)!) = 40320 / (24 × 24) = 70

Answer: 70

Problem 5: Find the value at row 5, column 1 in Pascal's Triangle.

Solution:

C(5, 1) = 5! / (1!(5 - 1)!) = 120 / (1 × 24) = 5

Answer: 5

Problem 6: Find the value at row 9, column 2 in Pascal's Triangle.

Solution:

C(9, 2) = 9! / (2!(9 - 2)!) = 362880 / (2 × 5040) = 36

Answer: 36

Problem 7: Find the value at row 10, column 5 in Pascal's Triangle.

Solution:

C(10, 5) = 10! / (5!(10 - 5)!) = 3628800 / (120 × 120) = 252

Answer: 252

Problem 8: Find the value at row 3, column 3 in Pascal's Triangle.

Solution:

The last element in each row is always 1. So, C(3, 3) = 1.

Answer: 1

Problem 9: Find the value at row 6, column 2 in Pascal's Triangle.

Solution:

C(6, 2) = 6! / (2!(6 - 2)!) = 720 / (2 × 24) = 15

Answer: 15

Problem 10: Find the value at row 11, column 3 in Pascal's Triangle.

Solution:

C(11, 3) = 11! / (3!(11 - 3)!) = 39916800 / (6 × 40320) = 165

Answer: 165

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