Write a recursive function called `productMatch` which takes an array of numbers and a target product and returns a boolean indicating whether any pair of distinct numbers matches the product when multiplied together.   Your function should accept a default argument called whichholds the return value. Each recursive step should reduce the size of thenumbers array after checking to see if pairs of numbers multiply to match thetarget product.

Modify the recursive Fibonacci function to employ the memoization technique discussed in this chapter. The function creates a dictionary and then defines a nested recursive helper function named memoizedFib. You will need to create a dictionary to cache the sum of the fib function. The base case of the fib function is the same as before. However, before making any recursive calls, the helper function looks up the value for the function’s current argument in the dictionary (use the method get, with None as the default value). If the value exists, the function returns it and skips any recursive calls. Otherwise, after the helper function adds the results of its two recursive calls, it saves the sum in the dictionary with the current argument of the function as the key. Note: The program should output in the following format: n      fib(n) 2       1 4       3 8       21 16      987 32      2178309 ---------------------------------------------------------------------------------- def…

Using a Java program solve the following problem using arrays: Past A: Coupon collector is a classic statistic problem with many practical applications. The problem is to pick objects from a set of objects repeatedly and determine how many picks are needed for all the objects to be picked at least once. A variation of the problem is to pick cards from a shuffled deck of 52 cards repeatedly and find out how many picks are needed before you see one of each suit. Assume a picked card is placed back in the deck before picking another. Write a program to simulate the number of picks needed to get total of four cards from each different suit and display the four cards picked (it is possible that a card may be picked twice). Here is a sample run of the program: Queen of Spades 5 of Clubs Queen of Hearts 4 of Diamonds Number of picks: 12 Sample run explanation: As you see in the above run, 12 picks are made to get the four cards from different suits. The other 8 picks (12-4-8) were from the…

1. Write a recursive method expFive(n) to compute y=5^n. For instance, if n is 0, y is 1. If n is 3, then y is 125. If n is 4, then y is 625. The recursive method cannot have loops. Then write a testing program to call the recursive method. If you run your program, the results should look like this: > run RecExpTest Enter a number: 3 125 >run RecExpTest Enter a number: 3125 2. For two integers m and n, their GCD(Greatest Common Divisor) can be computed by a recursive function. Write a recursive method gcd(m,n) to find their Greatest Common Divisor. Once m is 0, the function returns n. Once n is 0, the function returns m. If neither is 0, the function can recursively calculate the Greatest Common Divisor with two smaller parameters: One is n, the second one is m mod n. Although there are other approaches to calculate Greatest Common Divisor, please follow the instructions in this question, otherwise you will not get the credit. Meaning your code needs to follow the given algorithm. Then…