Step-by-step Solution:

The number of different binary trees that can be formed with \( n \) nodes is given by the Catalan number \( C_n \). The formula for the \( n^{th} \) Catalan number is given by:\[\] \(C_n = \frac{(2n)!}{(n+1)! \cdot n!} \) \[\] For ( n=6 ) , the number of different binary trees is given by \( C_6 \).\[\] Calculation of \( C_6 \): Using the formula: \(C_n = \frac{(2n)!}{(n+1)! \cdot n!}\) \[\] Plugging in ( n=6 ): \( C_6 = \frac {(2 \cdot 6)!}{(6+1)! \cdot 6!}\) \[\] \( C_6 = \frac{12!}{7! \cdot 6!}\) \[\] We calculate factorials: \[\] \( 12! = 479001600 \) \[\] - \( 7! = 5040 \) \[\] - \( 6! = 720 \)\[\] Substitute these into the formula: \( C_6 = \frac{479001600}{5040 \cdot 720}\) \( C_6 = 132 \) \[\] Correct Answer The number of different binary trees with 6 nodes is: 132 \[\] Correct Answer: (c)