The n-th row of Pascal's triangle has entries \binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}. Find every n for which the sum of this row lies strictly between 1000 and 3000, and show why the sum has the form it does.

The row sum is 2^n, so the only rows in range are n = 10 (sum 1024) and n = 11 (sum 2048).

Which Row of Pascal's Triangle Sums to a Given Range

Issue 24 Core Published 7 Oct 2026 By The Chalkboard Desk

The problem

The -th row of Pascal's triangle has entries . Find every for which the sum of this row lies strictly between and , and show why the sum has the form it does.

Rows are counted from , whose single entry is .
The bounds and are strict.

Take a minute

Reveal hint 1

There is a one-line answer for the row sum, and you can get it by expanding .

Reveal hint 2

Better still, count the subsets of an -element set in two different ways.

Solution

AnswerThe row sum is , so the only rows in range are (sum ) and (sum ).

Derivation

Identify the row sum.
Give the combinatorial reason.
Locate the candidates.
Check the boundaries.
Connect to a known row.