When question is "What is the number of ways to create a group of 5 from 17 girls and 20 boys when you have to select at least 3 girls
I needs a simple explanation why it is that $$ {17\choose3} {20\choose2} + {17\choose 4} {20 \choose 1} + {17 \choose 5} $$ (3 girls with 2 boys ... All girls) Which is the correct answer and I completely understand the logic
VS $$ {17 \choose 3} { 34 \choose 2} $$ (3 girls and random 2 from the remaining 34) Which is incorrect but I don't understand why it give a different number that the first