write function to output an arbitrary double number (which might be negativeusing only printdigit for / +allows statements of the form #include filename which reads filename and inserts its contents in place of the include statement include statements may be nestedin other wordsthe file filename may itself contain an include statementbutobviouslya file can' include itself in any chain write program that reads in file and outputs the file as modified by the include statements write recursive function that returns the number of in the binary representation of use the fact that this is equal to the number of in the representation of / plus if is odd write the routines with the following declarationsvoid permuteconst string str )void permuteconst string strint lowint high )the first routine is driver that calls the second and prints all the permutations of the characters in string str if str is "abc"then the strings that are output are abcacbbacbcacaband cba use recursion for the second routine prove the following formulasa log log(ab log evaluate following sumsthe = = = in  = estimate in/ what is (mod ) letfi be the fibonacci numbers as defined in section prove the followingn- = fi fn fn ph with ph ( )/  give precise closed-form expression for prove nthe following formulas = (    = =