Maths item of the month
Divisibility of consecutive integers
Let N = (n+1)(n+2)...(2n), i.e. the product of n consecutive integers from n+1 to 2n.
Prove that, for any positive integer n, N is divisible by 2n but not a higher power of 2.
For example, for n = 3:
N = 4×5×6
120 is divisible by 8 but not divisible by 16.
In the diagram various regular polygons, P, have been drawn whose sides are tangents to a circle, C.
Show that for any regular polygon drawn in this way:
Cones from a Circle
An angle θ is cut out of a circle of card to create two sectors: a major sector and a minor sector. The two sectors are then folded to make cones.
What angle θ is required to obtain the largest value for the sum of the volumes of the two cones?
An Unexpected Answer
Mr Student sets his class the following problem:
A committee of 3 students is to be chosen from a group of 13 students of which 8 are girls and 5 are boys. The students are selected at random, without replacement. What is the expected number of girls on the committee?
Anne Student immediately responds that the answer is .
She gives the reason that there are 3 students to be chosen and the proportion of girls is
so she calculated
Is she correct?
If the number of students was different would her method work?