Write the 4th row of Pascal’s Triangle. Find the element in the 5th row, 2nd column. Calculate the sum of the elements in the 3rd row.
Understand the Problem
The question presents concepts related to Pascal's Triangle and asks users to perform specific mathematical tasks related to its structure, including writing a row, finding a specific element, and calculating a sum of elements.
Answer
4th row: $1 \quad 4 \quad 6 \quad 4 \quad 1$; 5th row, 2nd column: $4$; Sum of 3rd row: $4$.
Answer for screen readers
The 4th row of Pascal’s Triangle is: $1 \quad 4 \quad 6 \quad 4 \quad 1$.
The element in the 5th row, 2nd column is: $4$.
The sum of the elements in the 3rd row is: $4$.
Steps to Solve
- Write the 4th Row of Pascal’s Triangle
Pascal's Triangle starts with rows indexed from 0. The rows are as follows:
- Row 0: $1$
- Row 1: $1 \quad 1$
- Row 2: $1 \quad 2 \quad 1$
- Row 3: $1 \quad 3 \quad 3 \quad 1$
- Row 4: $1 \quad 4 \quad 6 \quad 4 \quad 1$
So, the 4th row is $1 \quad 4 \quad 6 \quad 4 \quad 1$.
- Find the Element in the 5th Row, 2nd Column
To find an element in Pascal's Triangle, we use the combination formula:
$$ \binom{n}{r} = \frac{n!}{r!(n - r)!} $$
For the 5th row (which is indexed as row 4) and the 2nd column (which is indexed as column 1), we compute:
$$ \binom{4}{1} = \frac{4!}{1!(4 - 1)!} = \frac{4 \times 3 \times 2 \times 1}{1 \times (3 \times 2 \times 1)} = 4 $$
So, the element in the 5th row and 2nd column is $4$.
- Calculate the Sum of the Elements in the 3rd Row
The sum of the elements in the $n$-th row of Pascal's Triangle is given by $2^n$. Thus, for the 3rd row (indexed as row 2):
$$ \text{Sum} = 2^{2} = 4 $$
So, the sum of the elements in the 3rd row is $4$.
The 4th row of Pascal’s Triangle is: $1 \quad 4 \quad 6 \quad 4 \quad 1$.
The element in the 5th row, 2nd column is: $4$.
The sum of the elements in the 3rd row is: $4$.
More Information
Pascal's Triangle is a triangular array where each entry is the sum of the two entries directly above it. It is used in combinatorics for binomial coefficients and helps in calculating probabilities and algebraic expansions.
Tips
- Confusing the row and column indices. Remember that both are zero-indexed.
- Forgetting to use the combination formula correctly, especially with factorials.
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