The question states that the number of students with at least
questions wrong is
. Hence the number that actually get exactly
wrong,
, must be the number that get at least
wrong minus those that get at least
wrong. So we have
.
So if we let
be the total number of wrongly answered questions then
,
,
,
,
(letting
),
,
,
,
(letting
),
.
(I know I could have explicitly written out the sum, 'spot the pattern' - as I did on paper, but decided this was typographically easier than using an array).
So we have that
and solving gives us that
.