Permutation and combination algorithms

{Code ...} now there is an array that extracts 2 from the arr array (this may be 3 or more) Sub-arrays, put the length of the two arrays into the new array and add all the values in the new array. my current practice is to use loops. is there any formula that can be used directly...

$ Arr = ['A' => [,], 'B' => [,], 'c' =>, 9], 'D' => [6, 8, 1, 3, 5]; // a new array composed of the preceding values: $ new = [count ($ arr [a]) * count ($ arr [B]), count ($ arr [a]) * count ($ arr [c]), count ($ arr [a]) * count ($ arr [d]), count ($ arr [B) * count ($ arr [c]), count ($ arr [B) * count ($ arr [d]), count ($ arr [c) * count ($ arr [d])]; // Add all the values in the new array

Now there is an array that extracts 2 from the arr array (this may be 3 or more) Sub-arrays, put the length of the two arrays into the new array and add all the values in the new array. my current practice is to use loops. is there any formula that can be directly implemented?

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$ Arr = ['A' => [,], 'B' => [,], 'c' =>, 9], 'D' => [6, 8, 1, 3, 5]; // a new array composed of the preceding values: $ new = [count ($ arr [a]) * count ($ arr [B]), count ($ arr [a]) * count ($ arr [c]), count ($ arr [a]) * count ($ arr [d]), count ($ arr [B) * count ($ arr [c]), count ($ arr [B) * count ($ arr [d]), count ($ arr [c) * count ($ arr [d])]; // Add all the values in the new array

Now there is an array that extracts 2 from the arr array (this may be 3 or more) Sub-arrays, put the length of the two arrays into the new array and add all the values in the new array. my current practice is to use loops. is there any formula that can be directly implemented?

There is no formula for implementation. The formula is actually an algorithm, and the algorithm may have loops.

  [,], 'B' => [,], 'c' => [, 9], 'D' => [6, 8, 1, 3, 5]; $ arr_len = array (); foreach ($ arr as $ item) {$ arr_len [] = count ($ item) ;}// the length of the word group can be calculated for the above loop $ res = 0; // the result to be output $ c_arr_len = $ arr_len; foreach ($ arr_len as $ k1 => $ v1) {unset ($ c_arr_len [$ k1]); // Delete if (count ($ c_arr_len)> 0) {foreach ($ c_arr_len as $ k2 => $ v2) {$ res + = ($ v1 * $ v2) ;}} var_dump ($ res ); // get the result. You do not need to generate your $ new array.

Yes
2*(a1*a2 + a1*a3 + ... + a2*a3 ...) === ( a1 + a2 + a3 + ... )**2 - (a1**2 + a2**2 + ...)
Then, you can use map and reduce to avoid round robin.
However, the degree of parallelism remains unchanged, or n ^ 2.