Number of binary trees formed with 5 nodes are

The number of distinct binary trees that can be formed with 5 nodes can be determined using the formula given by the (n)th Catalan number. The (n)th Catalan number is given by:

[

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

]

For (n = 5), we plug in the values into the formula to get:

[

C_5 = frac{1}{5+1} binom{2*5}{5} = frac{1}{6} frac{10!}{5!5!}

]

Calculating this, we get:

[

C_5 = frac{1}{6} frac{3628800}{120 times 120} = frac{1}{6} frac{3628800}{14400}

]

[

C_5 = frac{1}{6} times 252 = 42

]

Therefore, the number of distinct binary trees that can be formed with 5 nodes is 42.