Two teachers cheat with grades!

Probability Level 5

Class 9a has 20 pupils and the average of their grades is 3.3 , their individual results are currently shrouded in mystery.

Class 9b consists of 25 pupils, there is 1 people who got a 5, and for every grade left, namely 1,2,3 and 4 are 6 pupils. Therefore the average of their grades is 2.6

Now their teachers want to improve the grades of both classes by making 2 or 3 pupils switch from class 9b to 9a.

Now how many possible groups of 2 and 3 pupils from class 9b can be formed, so that the average grade of both classes improve?

Note: 1 is the best possible grade, grade 6 the worst. Therefore an average grade of 3.0 for Class 9a would be an improvement. Possible grades are the integers: 1,2,3,4,5,6

The answer is 866.

1 solution

Gary Hu
Jul 23, 2015

In order for the average grades of both classes to improve, the average grade of the 2 or 3 students moved must be greater than 2.6 and less than 3.3. Three cases are possible:

Case 1: Two students moved with average grade of 3 (sum of grades = 6)

Possible solutions:

$(1,5)\rightarrow 6 \times 1=6$ combinations

$(2,4)\rightarrow 6 \times 6=36$ combinations

$(3,3)\rightarrow {6 \choose 2}=15$ combinations

Case 2: Three students moved with average grade of $\frac{8}{3}$ (sum of grades = 8)

Possible solutions:

$(1,2,5) \rightarrow 6 \times 6 \times 1=36$ combinations

$(1,3,4) \rightarrow 6 \times 6 \times 6=216$ combinations

$(2,2,4) \rightarrow {6 \choose 2} \times 6=90$ combinations

$(2,3,3) \rightarrow 6 \times {6 \choose 2}=90$ combinations

Case 3: Three students moved with average grade of 3 (sum of grades = 9)

Possible solutions:

$(1,3,5) \rightarrow 6 \times 6 \times 1=36$ combinations

$(1,4,4) \rightarrow 6 \times {6 \choose 2}=90$ combinations

$(2,2,5) \rightarrow {6 \choose 2} \times 1=15$ combinations

$(2,3,4) \rightarrow 6 \times 6 \times 6=216$ combinations

$(3,3,3) \rightarrow {6 \choose 3}=20$ combinations

$6+36+15+36+216+90+90+36+90+15+216+20=\boxed{866}$

Moderator note:

This looks like a pretty tedious approach. Is there another way which allows us to avoid all of these cases?

Nicely done :)

Alisa Meier - 5 years, 10 months ago

I agree with you alisa

Vadarai Ingram - 3 years ago

Two teachers cheat with grades!

The answer is 866.

1 solution

Moderator note:

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