1
$\begingroup$

There are 229 girls and 271 boys at a school. They are divided into 10 groups of 50 students each, with numbering 1 to 50 in each group. A quartet consists of4 students from 2 different groups so that there are two pairs of students having identical numbers. Show that the number of quartets with an odd number of girls is itself odd. Any help, please? Would it help to extend to Mod 4? Is the total number of quartets $${50 \choose 2}\cdot {10 \choose 2} $$???

$\endgroup$
2
  • $\begingroup$ I'm assuming we need to fit every student into a quartet? IE we need to form 125 quartets? because if not, the statement is not true. $\endgroup$ Commented Apr 5, 2014 at 0:36
  • $\begingroup$ My interpretation was that every student goes into a quartet but possibly more than one. $\endgroup$
    – user140519
    Commented Apr 5, 2014 at 1:11

1 Answer 1

2
$\begingroup$

It looks to me that every student goes into a quartet. If so, there are an even number of girls in quartets with an even number of girls. As there are an odd number of girls total...

$\endgroup$
2
  • $\begingroup$ Sorry I can't comment but can but can Ross Millikan please elaborate. $\endgroup$
    – Dominic
    Commented Apr 5, 2014 at 8:31
  • $\begingroup$ @Dominic: If all the girls go into quartets, and there are an odd number of girls, the number of girls in quartets with an odd number of girls must be odd. This is because even*odd=even, so if there were an even number of quartets with an odd number of girls, the total number would be even. $\endgroup$ Commented Apr 5, 2014 at 14:43

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .