Question

in pascal’s triangle, which row sums to 128?
a. fifth row
b. seventh row
c. sixth row
d. eighth row

Explanation:

Step1: Recall the formula for the sum of a row in Pascal's Triangle

The sum of the elements in the \( n \)-th row of Pascal's Triangle is \( 2^n \), where the first row (row 0) is \( [1] \) with sum \( 2^0 = 1 \).

Step2: Set up the equation to find \( n \)

We need to find \( n \) such that \( 2^n = 128 \). We know that \( 2^7 = 128 \) (since \( 2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64, 2^7 = 128 \)). But we need to be careful with the row numbering. If we consider row 0 as the first row (sum \( 1 = 2^0 \)), then the row number \( k \) has sum \( 2^k \). Wait, actually, the \( n \)-th row (starting from row 0) has sum \( 2^n \). So if the sum is 128, \( 2^n = 128 \implies n = 7 \). But in the options, the rows are labeled as fifth, seventh, sixth, eighth. Wait, maybe the row numbering here starts from row 1. Let's check:

• Row 1 (first row): elements \( [1, 1] \), sum \( 2 = 2^1 \)
• Row 2: \( [1, 2, 1] \), sum \( 4 = 2^2 \)

-...

• Row \( k \): sum \( 2^k \)

So if the sum is 128, \( 2^k = 128 \implies k = 7 \). So the seventh row (since row number is \( k \)) has sum 128.

Answer:

b. Seventh row