Thus you get
c) Let m be the number of men chosen, the corresponding number of women must then be 10-m; to satisfy the condition that m<10-m, m can only assume the values m=0, 1, 2, 3, or 4. Hence, by summing all these cases we get
As to the number of strings of digits (not numbers) of length 6 that contain exactly 3 times the digit 4: you can choose the position of the 3 digits that are 4 in ways, and each of these can be combined with one of the ways to choose the remaining 2 digits, different from 4.
Thus you get possibilities.
Note that there is a difference depending on whether we are talking here of the number of strings of length 6 or of numbers with 6 digits, because in the latter case, the first digit would not be allowed to be 0.