**Shortcut 91:**

Events

* Dependent=multiplication*

* Independent =addition*

Question:

There 3 busses from city A to city B and there are 5 busses from city B to city C. In how many ways a person can travel from city A to C through B?

Answer:

Choosing a bus from city B depends on choosing a bus from city A.

Number of ways = 3 x 5 = 15

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Question:

There 3 busses from city A to city B and there are 5 busses from city A to city C. In how many ways a person can travel from city A to C or B?

Answer:

Choosing a bus to city B or C are not dependent.

Number of ways = 3 + 5 = 8

**Shortcut 92:**

Arrangement with repetition

[latex]n^n;n^r[/latex]

Question:

In how many ways the letters of the word “ORANGE” can be arranged with repetition?

Answer:

n = 6 (n = total number of elements)

Since all the elements are taken,

Number of arrangements = 66

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Question:

In how many ways three letters from the word “ORANGE” can be arranged with repetition?

Answer:

n = 6; r = 3 (r = number of elements taken for arrangement)

Number of arrangements = 63 = 216

**Shortcut 93:**

Arrangement without repetition

* [latex]n!;nP_r=n!/(n-r)![/latex]*

Question:

In how many ways the letters of the word “MANGO” can be arranged without repetition?

Answer:

n = 5

Since all the elements are taken for arrangement,

Number of elements = n! = 5! = 120

Question:

In how many ways any three letters of the word ‘MANGO” can be arranged without repetition?

Answer:

n= 5; r = 3

nPr = 5!/(5 – 3)! = 120/2 = 60 ways

**Shortcut 94:**

Elements occurring together

* 2!*(n-1)*

* 3!*(n-2) *

Question:

In how many ways letters of the word “ORANGE” arranged so that the vowels will always occur together?

Answer:

n = 6

Three letters ‘O, A and E’ should occur together.

Since three letters occur together

3! x (6 – 2)! = 6 x 24 = 144

Note:

If there are 4 letters occurring together, then

4!(6 – 3)!

If there are 5 letters occurring together, then

5!(6 – 4)!