Combinations from selecting four numbers from the set of the first eight positive integers

Question

Four numbers are to be selected from the set of the first eight positive integers. Find how many possible combinations there are if: there is atleast one odd number

Now, you could do this the manual hard way taking cases so (4C1*4C3) + (4C2*4C2) + (4C3*4C1) = 68 Or you could calculate the possible combinations for all numbers to be even: Thats' clearly only one combination Then subtract these from all possible combinations without restriction 8C4 -1 = 70-1 = 69

What sily thing am I doing wrong here? Why do the answers vary?

Answer 1

In the first version, you have forgotten the case where all four numbers are odd.