Hello, ananya jana!
I'll take a guess at what the question asks . . .
Each roll has 6 possible outcomes.A die is rolled 10 times.
Find the number of ways so that outcomes always contain a 1, 2, or 3.
With a sequence of 10 rolls, there are: . possible outcomes.
How many of these rolls contain no 1s, 2s, or 3s?
The rolls must consist of 4, 5, and/or 6 only.
. . There are: . ways.
Therefore, there are: . ways.
Hello again, ananya jana!
See? . . . You have a different interpretation of the problem!
If that was the exact wording, the problem is poorly written.
A die is rolled 10 times.
Find the number of ways so that the outcomes
always contain at least one 1, one 2, and one 3.
There are: . possible outcomes.
Let: .
. .
. .
. .
Formula: .
. .
. .
There are 26,210,196 outcomes with no 1 or no 2 or no 3.
Therefore, there are: . outcomes
. . that contain at least one 1, one 2, and one 3.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Another interpretation . . .
Number of outcomes that contain exactly one 1, one 2, and one 3.
The 1,2,3 can occur in any of: . sequences of rolls.
The other seven rolls must not be 1, 2, or 3: . ways.
Therefore, there are: . ways.