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

.