Problem 3
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A Pascal-like triangle has $10$ as the top row and $10$ followed by $1$ as the second row. In each subsequent row the first number is $10$, the last number is $1$, and, as in the standard Pascal Triangle, each other number in the row is the sum of the two numbers directly above it. The first four rows are shown below.
$$10$$
$$10, 1$$
$$10, 11, 1$$
$$10, 21, 12, 1$$
What is the sum of the digits of the sum of the numbers in the $11$th row?
$$\textbf{(A) } 11 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 14 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17$$
Difficulty: 27/100 — Heavily dependent on pattern recognition.
Core Concepts: pattern recognition, arithmetic
Challenges: Failure to recognize the pattern can lead to an excessive bash.
Techniques: Being able to recognize a pattern by computing early sums. Taking the remainder when divided by 9 offers a quick solution as well.
Error-prone Steps: Assuming the same sum as a normal Pascal Triangle, incorrectly determining the row sum by applying the pattern inaccurately.
Ideal Time:
Experienced: ≤ 30-60 sec
Intermediate: ≤ 1-2 min
Beginner: ≤ 3-4 min