Tips & Challenges

Dividing people or distinct objects into groups

Calculating number of ways of dividing people or distinct objects into groups of the same size often pose difficulty to students.

Hopefully these examples will help students understand the concept.

Basic Example

How many ways can 10 children be divided into two groups of 5?

1

When dividing people or distinct objects into groups, if p groups are indistinguishable, we need to divide by p! to avoid double counting.

Challenging example 1 

At a particular reception, 9 guests are to stand at 3 identical round tables. How many ways can this be done if there are at least two people at each table?

3

Challenging example 2 (SAJC 2011/MYCT/2/6)

2

Differentiation chain rule

Chain rule is very useful when we need to differentiate complex functions or doing implicit differentiation.

differentiation 1

differentiation 2

Distinguish X1+X2 from 2X

This challenging question on normal distribution was posted in Edusnap by a student.

Screenshot_2015-05-27-13-52-49

There were two mistakes made:

1) Interpreting the distribution of 2 large hampers and n small hampers as 2L + nS.

2) Calculation of variance

Here is the correct solution:

hamper

1

2

Smallest possible value of n = 9

————————-

When the question mentions n items, the distribution is X1+X2+…Xn.  When the question asks for “n times of a randomly chosen..”, then the distribution is nX.

Students also need to be familiar with calculating the expectation and variance of the sum or difference of normal distribution using the formula:

form

Challenging Probability problem 5

An unbiased die is thrown 6 times. Calculate the probabilities that the six scores obtained will

i) be 1, 2,3, 4, 5,6 in some order

ii) have a product which is an even number

iii) consists of exactly two 6’s and four odd numbers

Solution

dice prob

iii Alternative solution

6C2: 6 slots choose 2 slots to insert the two 6s. Remaining 4 slots automatically will be for the four odd numbers.
3^4: Each odd number has 3 choices. for 4 odd numbers will be 3^4
Probability = (6C2 x 3^4) / 6^6 = 5/192

 

Calculating Mean and Variance of coded data

coded-data

Venn diagram, Inclusion and Exclusion Principle

Venn diagram, Inclusion and exclusion principle can be useful in solving permutations, combinations and probability problems.

It is especially useful in solving combinations where the cases are not mutually exclusive and hence addition principle cannot be applied.

Example 1

venn diagram pnc

venn diagram p2