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2024 AMC 10B Problem 1

2024 AMC 10B Problem 1AMC 10 Step by Step

Name: Date: September 13, 2026

Basic Counting · difficulty 1 of 5 · about 1 min · position P1-10

In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?

A)
$2021$
B)
$2022$
C)
$2023$
D)
$2024$
E)
$2025$

Answer

Key insight

Position from the left plus position from the right counts the same person twice, so the line has 1013 + 1010 - 1 people.

Solution

Being $1013$ th from the left means there are $1012$ people to this person's left. Being $1010$ th from the right means there are $1009$ people to the right.

The line consists of those two groups plus the person in question:
$1012 + 1 + 1009 = 2022.$

The answer is $\boxed{\textbf{(B)}\ 2022}$ .

Why this works

" $k$ th from one end" always means $k - 1$ people are on that side. Counting the two sides separately and adding the person back avoids the double count that comes from simply adding the two positions. A tiny example ( $4$ th from the left and $2$ nd from the right in a line of $5$ ) confirms the rule "left position plus right position minus one."

The trap

Adding 1013 + 1010 = 2023 without subtracting the person who was counted twice, which is choice (C).

Common mistakes

Adding 1013 + 1010 = 2023 without subtracting the person who was counted twice, which is choice (C).
Subtracting twice (once for each side) and getting $2021$ .

Techniques

Set up the equation/formula and compute; no special trick needed · Compute small cases, spot the pattern, generalize