A soccer team has 3 goalkeepers, 12 defenders, 10 midfielders, and 8 forwards. The coach has planned a 4-3-3 formation for each game of the season, meaning that only 4 defenders, 3 midfielders, 3 forwards, and 1 goalie will be on the field. How many starting lineups can the coach designate?
Consider that the defenders, midfielders, and forwards have assigned positions, that is, leftback/right-back/left-central/right-central for defenders, and left/central/right for midfielders and forwards
So, to solve that problem, I applied the permutation formula n!/(r!(n-r)!) to each set of positions (goalkeeper, midfields, defenders, and forwards) since each player is assigned to a distinct position. For example, for the defenders I got the following (12!)/(4!8!) = 495 (same for other set of positions). Then, I applied multiplication rule and multiplied the result of each set to each other which gave me the following:
495 * 210 * 56 * 3 = 17,463,600.
I am just not sure if it's a correct way because the positions could be different and I am not sure if it affects the formula. Would really appreciate any response and help! Thanks.
Apparently, you have used the combination formula, although there is either a typo or an error in one of the terms (should be $120$, not $210$)
But apart from that, note carefully, that specific positions are also to be assigned to each category, so you need to further multiply by $4!3!3!$