Dice problems agaaaain

The possible outcomes of a single die can be represented using the polynomial: $x+{ x }^{ 2 }+{ x }^{ 3 }+{ x }^{ 4 }+{ x }^{ 5 }+{ x }^{ 6 }$

Then the possible outcomes of three dice is representable as the third power of this polynomial:

$x^{18} + 3 x^{17} + 6 x^{16} + 10 x^{15} + 15 x^{14} + 21 x^{13} + 25 x^{12} + 27 x^{11} + 27 x^{10} + 25 x^{9} + 21 x^{8} + 15 x^{7} + 10 x^{6} + 6 x^{5} + 3 x^{4} + x^{3}$

There's one way of getting 18, there are three ways of getting 17, and so on. The total number of ways the dice can appear is the sum of the coefficients, 216. There are 27 ways of getting 10.

The possible combinations of $3$ integers, each between $1$ and $6$ inclusive, which sum to 10, along with the number of possible permutations, are as follows:

(i) $1, 3, 6 \Longrightarrow 6$ permutations,

(ii) $1, 4, 5 \Longrightarrow 6$ permutations,

(iii) $2, 2, 6 \Longrightarrow 3$ permutations,

(iv) $2, 3, 5 \Longrightarrow 6$ permutations,

(v) $2, 4, 4 \Longrightarrow 3$ permutations,

(vi) $3, 3, 4 \Longrightarrow 3$ permutations.

This gives us a total of $27$ outcomes out of the $6^{3} = 216$ outcomes that can occur without restrictions. Thus the desired probability is $\frac{27}{216} = \boxed{\frac{1}{8}}$ .

Here is the list of permutation for getting a total of 10
6 3 1 == 6
6 2 2 == 3
5 4 1 == 6
5 3 2 == 6
4 4 2 == 3
4 3 3 == 3
Total == 27
So, Required probability = 27/216 = 1/8

Just count the permutations pivoting on one die when it's 1 the other two ought to be 9 there are 4/36 permutations for this.

when it's 2 the other two ought to be 8 - 5/36 permutations

when it's 3 the other two ought to be 7 - 6/36 permutations

when it's 4 the other two ought to be 6 - 5/36 permutations

when it's 5 the other two ought to be 5 - 4/36 permutations

when it's 6 the other two ought to be 4 - 3/36 permutations

By probability multiplication rule and the fact that on the pivoted dice each number has 1/6 to roll. we multiply all of the other two dice probabilities times 1/6 and then add up 27/216 = 1/8

Di ak maaram. Chamba la adto

guesss..........any easier methods plz.....

event is 27 and sample space is 6^3 so P= 27/6^3 or 1/8 which is the answer XD