Probability MCQ3: (No.of ways for Arrangement)

Probability MCQ3: What are the chances that no two boys are sitting together for a photograph if there are 5 girls and 2 boys?

a) 1/21                    b) 4/7                     c) 2/7                 d) 5/7

Solution:

Keep in mind that :

If there are n things, then number of ways to arrange them  = n!

Where,

\[ n!=n \times (n-1) \times (n-2) \times (n-3) —– so – on \]

( we read n! as “n factorial” —a mathematical terminology for a number n !)

Let us first understand factorial concept  by examples;

What is 4! ?

Ans: ​\( 4!=4\times 3\times 2\times1=24 \)​

What is 6 !?

ans  :\( 6!=6\times 5 \times 4 \times 3 \times2 \times1=720 \)

Now lets come to the question:

Number of girls =5

Number of boys =2

Total=5+2=7

so, number of ways to arrange 7 persons =7! = 7x6x5x4x3x2x1=5040

i-e we can arrange 7 persons in different 5040 ways !

If lets suppose two boys sit together – we take their pair as one person ! – then number of total persons will be= 5+1=6

Then number of ways to arrange 6 persons = 6! = 6x5x4x3x2x1=720

or

number of ways to arrange  5 girls  and one pair of boys=720

Since together sitting boys (pair of boys) can also interchange their positions – i-e one boy sitting right side  of the other can sit left as well ! — two ways of sitting together- left or right side!

So, the number of ways doubles in this case !

i-e number of ways to arrange 5 girls and two together sitting boys= 2x 720=1440

Now ,

Number of ways of arrangement that no two boys sit together =Total  Number of  ways of arrangement of 7 person – The number of ways in which two boys sit together = 5040  –  1440 = 3600

We know for Probability of an ‘event’,  formula in statistics is;

\[ P(Event)=\frac {Event}{TotalSample} \]

Here the event is “number of ways boys not sit together” – and total sample is “number of ways of arranging 7 person”

So ,

if  we suppose,

E=number of ways boys not sit together

and

T=number of ways of arranging 7 person

then,

\[ P(E)=\frac {3600}{5040}=\frac {5}{7} \]

So the option   ‘d’ is correct !

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