I have to choose any seven numbers from the numbers 1 through number 49 inclusive. How many combinations are possible?

Answer 1

Welcome to the world of permutations. We have to sort out one critical issue before we forge ahead. Is this a replacement or non-replacement set that we are drawing our numbers from? The question said "any seven numbers" and was no more specific than that. In conbinatronic permutations, we generally mean a non-replacement set as opposed to a replacement set. But the question said "any seven numbers" and that could mean a replacement set. What we are talking about is whether when we take a number from the number pool, do we put it back so it may possibly be chosen again or do we not? Replacement is when we put it back so it might be drawn again, and non-replacement is when it can be used only once in a draw. In a non-replacement set, we have to reduce the pool by one number after each time we choose an element. If out problem was this non-replacement kind, the number of possible combinations for the seven digits would be 49 x 48 x 47 x 46 x 45 x 44 x 43 = 432,938,943,360 That's 432 billion, 938 million, 943 thousand, 360. In a replacement set, the possible number combinations for seven digits from a pool of from 1 through 49 is found by multiplying the number of elements in the pool by itself as many times as the number of elements taken from the pool. In our problem, 49 x 49 x 49 x 49 x 49 x 49 x 49 = 497 = 623,987,825,041 That's 623 billion, 987 million, 825 thousand, 041.