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Three dice are rolled independently. What is the probability that the highest and the lowest value differ by $4$?

1. $\left(\dfrac{1}{3}\right)$
2. $\left(\dfrac{1}{6}\right)$
3. $\left(\dfrac{1}{9}\right)$
4. $\left(\dfrac{5}{18}\right)$
5. $\left(\dfrac{2}{9}\right)$

2/9 is correct.

Its a beautiful question and indeed a beautiful solution @aritra nayak

When should we count the permutations, and when not count them?

Case 1: largest is $5$, smallest $1$ and middle is $2$ or $3$ or $4 : 3\times 3!$

Case 2: largest is $5$, smallest $1$ and middle is $1$ or $5 : \dfrac{3!\times 2}{2!}$

Case 3: largest is $6$, smallest $2$ and middle is $3$ or $4$ or $5 : 3\times 3!$

Case 4: largest is $6$, smallest $2$ and middle is $6$ or $2: \dfrac{3!\times 2}{2!}$

So, probability the highest and the lowest value differ by $4,$

$\quad=\dfrac{\left( 3\times 3!+\dfrac{3!\times 2}{2!}+3\times 3!+\dfrac{3!\times 2}{2!}\right)}{6^{3}} =\dfrac{2}{9}.$

Correct Option: E

How you got 3 * 3! ? and (3! * 2 ) / 2! ? Can you please explain ...
@shreya would you please explain how you got the 2nd case and 4th....
For case 1:  (1, 2/3/4 ,5) for middle place we have 3 choice and all places can be arranged in 3! ways

So total is 3*3! ways,

For case 2: (1, 1/5 ,5) for middle place we have 2 choice and all places can be arranged in 3!/2! ways

So total is 2*(3!/2!) ways,
 Sno Lowest Value Highest Value Values at dice No of ways 1 1 5 {1,1,5} 3 2 1 5 {1,2,5} 6 3 1 5 {1,3,5} 6 4 1 5 {1,4,5} 6 5 1 5 {1,5,5} 3 6 2 6 {2,2,6} 3 7 2 6 {2,3,6} 6 8 2 6 {2,4,6} 6 9 2 6 {2,5,6} 6 10 2 6 {2,6,6} 3

By naive defn of Prob = # outcomes which satisfy our constraint / total # of outcomes

Prob = 48 / 63 = 2 / 9

## The correct answer is (E) 2/9

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### 1 comment

Plain and simple approach but I guess thinking and executing this will consume atleast 5 mins in Gate

suppose three dices are there A,B,C, three are rolled independently, see we get 4 difference between max and min when we get either (1,5) or (2,6) in any of the two dices. okay?

1st case: we get (1,5), 1 and 5 can be the result of any of the two dice in 3C2=3 ways, either (A-B) or (B-C) and (C-A) and also they can be permutated between them,so 3C2*2! ways possible, now for each way 3rd result can come either 1 or 2 or 3 or 4 or 5(6 cant be possible as min:1 and max:5) But we cant do 3C2*2!*5 because then (1,1,5), (1,5,1),(5,1,1),(1,5,5),(5,1,5),(5,5,1) all these 6 ways are counted two times.So we do like below:
When 3rd result is 1, no of ways=3!/2!=3  because its no of permutation of 1,1 and 5

When 3rd result is 5,no of ways=3!/2!=3 because its no of permutation of 1,5 and 5

when 3rd result is 2 or 3 or 4, 3C2*2!*3=18 ways possible.so totally 18+3+3=24

2nd case: we get (2,6), 2 and 6 can be the result of any of the two dice in 3C2=3 ways, either (A-B) or (B-C) and (C-A) and also they can permutate between them,so 3C2*2! ways possible, now for each way 3rd dice result can come either 2 or 3 or 4 or 5 or 6(min:2 and max:6) But we cant do 3C2*2!*5, So we do like below:

Similarly as shown above, when 3rd result is 2 or 6,no of ways=2*3!/2=6 ways possible,when 3rd result is 3 or 4 or 5,3C2*2!*3 =18 ways possible.So totally 18+6=24 ways possible.

Finally total=24+24=48 ways possible
sample space= 6^3=216
so probability= 48/216=2/9

Correct Answer: $E$

It becomes simple the moment we enumerate all favourable ways

Case 1: Minimum value on dice is 1 and the maximum value on dice is 5.

 Numbers on 3 dices Total number of ways these 3 numbers can appear on the dice 1,1,5 $\frac{3!}{2!}=3$ 1,2,5 $3!=6$ 1,3,5 $3!=6$ 1,4,5 $3!=6$ 1,5, $\frac{3!}{2!}=3$

Total favourable ways in this case=3+6+6+6+3=24.

Case 2: Minimum value on the dice is 2 and the maximum value is 6

Now when 2 and 6 have appeared the third number can be anyone from $\{2,3,4,5,6\}$ so that the difference between the minimum and the maximum number remains 4.

Now, this case is symmetrical to case 1 where there are two possibilities when the number on the third dice has already appeared in one of the two dices and remaining 3 possibilities when the number on three dices are all distinct and hence this case also has favourable ways=24.

Total favourable ways=48.

Total possible outcomes=$6^3$

Probability=$\frac{48}{216}=\frac{2}{9}$

nice ayush
nice aritra nayak
Thanks Ayush sir and  aritra sir

There will be 2 possibilities for highest and lowest differ by 4, i.e (1,5),(2,6) and they can arrange in 2! ways

Now among 3 dies 2 dice can be selected in 3C2 ways

Case 1:

for (1,5) the 3rd dice value can be any value from 1 to 5, but it cannot be 6, Because in that case difference cannot be 4

so probability is 3C2*2!*5/ (6^3)

Case 2:

for (2,6) the 3rd dice value can be any value from 2 to 6, but it cannot be 1, Because in that case difference cannot be 4

so probability is 3C2*2!*5/ (6^3)

So,  probability that the highest and the lowest value differ by 4

=3C2*2!*5/ (6^3)+3C2*2!*5/ (6^3)= 5/18

Ans is (D)

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