Question

1. Create a scenario that can be used to model a counting question (permutation, combination or neither.)

2. Create a scenario that can be used to model a counting question (permutation, combination or neither.)

3. Create a scenario that can be used to model a counting question with multiple 'not', 'and', 'or' clauses

Answer 1

Permutation:

The repetation is allowed in permutation that means the order is not important.

For example, if we have two elements x and y, then there are two possible arrangements, xy and yx.

The formula for permutation gives the number of arrangement,that will give in how many ways we can arrange r objects out of n objects with repetation.

Combination:

The repetation is not allowed in combination that means the order is important.It is nothing but a selection.

For example, if we have two elements x and y, then there is one possible arrangements, either xy or yx

The formula for combination gives the number of selection,that will give in how many ways we can select r objects out of n objects without repetion.

Example:

Q1 : How many words can be formed by using 4 letters from the word “CAUTION” ?
Solution : The word “CAUTION” has 7 different words.There is no repetaion of letters.
Therefore, required number of words = ⁷ P ₄ = 7! / (7 – 4)!
=> Required number of words = 7! / 3! = 7*6*5*4*3! / 3! = 7*6*5*4 =840

Q2 : How many words can be formed by using the letters from the word “DRIVER” such that all the vowels are always together ?
Solution : In these type of questions, we assume all the vowels to be a single character, i.e., “IE” is a single character.
So, now we have a total of 5 characters in the word, namely, D, R, V, R, IE.
But, R occurs 2 times.
Number of possible arrangements = 5! / 2! = 60
Now, the two vowels can be arranged in 2! = 2 ways.
=> Total number of possible words such that the vowels are always together= 60 x 2 = 120

Q3: In how many ways can a team of 11 cricketers be chosen from 9 batsman and 6 bollowers to give majority of batsman. If atleast 3 bollowers are to included.

Solution:

We have to choose team of 11, from 9 batsman in which at least 3 are bollowers and there should be majority of batsman.

It can behappens when i) 3 bollowers, 8 batsman

                                      ii) 4 bollowers, 7 batsman

                                      iii) 5 bollowers, 6 batsman'

The next case can not be happen since there would be more batsman than bollowers.

Thus, the number of ways to select a team would be ,

Thus, in 918 ways we can select a team of 11 with given condition.