Recipe
Apply the Catalan number formula
Use when: you recognize a Catalan-number pattern, a lattice path that never crosses a diagonal, a balanced parenthesis string, or a binary tree shape, and need the count for a specific size
Goal: the correct Catalan number C_n, with n correctly identified from the problem
Taught in Catalan numbers
Quick recipe
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1
Match the problem's size parameter, path length, number of parenthesis pairs, or number of tree nodes, to the theorem's n.
Example
5 pairs of balanced parentheses correspond to n=5.
Why: the formula only gives the right number once n is read off correctly; miscounting n by one is the single most common slip.
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2
Substitute the value of n from Step 1 into C_n = \frac{1}{n+1}\binom{2n}{n}.
Example
C_5 = \frac{1}{6}\binom{10}{5}.
Why: writing the substituted formula before computing anything catches a setup error, such as using the wrong n, before it becomes an arithmetic mistake.
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3
Evaluate \binom{2n}{n}.
Example
\binom{10}{5}=252.
Why: this single number is what the rest of the computation depends on.
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4
Divide the binomial coefficient from Step 3 by n+1; the result is always a whole number.
Example
252 / 6 = 42.
Why: C_n is always an integer despite the fraction sitting in front of the formula, so a non-integer result is a signal that something upstream went wrong.
Common mistakes
- Setting n to the problem's raw count (such as "8 parentheses") instead of the number of pairs or steps it actually represents, off by a factor of 2.
- Computing \binom{2n}{n} but forgetting to divide by n+1, reporting a much larger number than the true Catalan number.
- Getting a non-integer after dividing, which should be treated as a red flag that n or the binomial coefficient was set up wrong, not rounded away.
- Confusing C_n's index with a row number in Pascal's triangle when reading values off a table.