CUET PG 2021 — Computer PYQ

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Question

CUET PG 2021 — Computer PYQ

CUET PG | Computer | 2021

The number of different binary trees with 6 nodes is

Choose the correct answer:

Explanation

1. The Catalan Equation

The number of distinct binary trees $T$ with $n$ nodes is given by the $n^{th}$ Catalan number:

T(n) = C_n = \frac{1}{n+1} \binom{2n}{n}

Alternatively, it can be written as:

C_n = \frac{(2n)!}{(n+1)! \, n!}

2. Step-by-Step Calculation for $n = 6$

Substitute $n = 6$ into the formula:

C_6 = \frac{1}{6+1} \binom{2 \times 6}{6}

C_6 = \frac{1}{7} \binom{12}{6}

Calculating the Binomial Coefficient $\binom{12}{6}$ :

\binom{12}{6} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1}

\binom{12}{6} = 924

Final Calculation:

C_6 = \frac{924}{7}

\mathbf{C_6 = 132}