Find the number of valid parentheses expressions of given length | Returns value of Binomial Coefficient C ( n , k ) ; Since C ( n , k ) = C ( n , n - k ) ; Calculate value of [ n * ( n - 1 ) * -- - * ( n - k + 1 ) ] / [ k * ( k - 1 ) * -- - * 1 ] ; A Binomial coefficient based function to find nth catalan number in O ( n ) time ; Calculate value of 2 nCn ; return 2 nCn / ( n + 1 ) ; Function to find possible ways to put balanced parenthesis in an expression of length n ; If n is odd , not possible to create any valid parentheses ; Otherwise return n / 2 'th  Catalan Number ; Driver Code
< ? php function binomialCoeff ( $ n , $ k ) { $ res = 1 ; if ( $ k > $ n - $ k ) $ k = $ n - $ k ; for ( $ i = 0 ; $ i < $ k ; ++ $ i ) { $ res *= ( $ n - $ i ) ; $ res /= ( $ i + 1 ) ; } return $ res ; } function catalan ( $ n ) { $ c = binomialCoeff ( 2 * $ n , $ n ) ; return $ c / ( $ n + 1 ) ; } function findWays ( $ n ) { if ( $ n & 1 ) return 0 ; return catalan ( $ n / 2 ) ; } $ n = 6 ; echo " Total ▁ possible ▁ expressions ▁ of ▁ length ▁ " , $ n , " ▁ is ▁ " , findWays ( 6 ) ; ? >