Sherlock Counts Ways

Watson gives an array of colored balls to Sherlock. All balls are identical in shape and size and hence, two balls with same color are indistinguishable. He gives to Sherlock an integer K and wants Sherlock to choose a bundle of K balls from the array. Sherlock is intrigued by the problem and starts wondering in how many different ways he can choose K such balls. Two ways are different if for any