Question

The number of ways of arranging the digits 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, and 6 taken all at a time so that the 3s are separated is:

a. 120
b. 240
c. 360
d. 480

Answer 1

To calculate the number of ways of arranging the given digits so that the 3s are separated, we need to consider three cases. For each case, we can use permutations to calculate the number of arrangements. The total number of arrangements is 100.

Step-by-step explanation:

To calculate the number of ways of arranging the given digits so that the 3s are separated, we need to consider three cases:

1. Arranging the digits 1, 1, 1, 1, 1, 2, 2, 2, 4, 5, 5, 6 (where the 3s are separated).
2. Arranging the digits 1, 1, 1, 1, 1, 2, 2, 2, 3, 5, 5, 6 (where the first two 3s are adjacent).
3. Arranging the digits 1, 1, 1, 1, 1, 2, 2, 2, 5, 3, 5, 6 (where the last two 3s are adjacent).

We can calculate the number of arrangements for each case by using permutations. For the first case, we have 5 repetitions of the digit 1, 3 repetitions of the digit 2, and individual occurrences of the digits 4, 5, and 6. Hence, the number of arrangements is given by:

Number of arrangements = (5! / (3!)) * 3! = 10 * 6 = 60

Similarly, for the second case, we have 5 repetitions of the digit 1, 3 repetitions of the digit 2, 2 repetitions of the digit 3, and individual occurrences of the digits 5 and 6. Hence, the number of arrangements is given by:

Number of arrangements = (5! / (3! * 2!)) * 2! = 10 * 2 = 20

Finally, for the third case, we have 5 repetitions of the digit 1, 3 repetitions of the digit 2, 2 repetitions of the digit 5, and individual occurrences of the digits 3 and 6. Hence, the number of arrangements is given by:

Number of arrangements = (5! / (3! * 2!)) * 2! = 10 * 2 = 20

Adding up the number of arrangements from each case, we get a total of 60 + 20 + 20 = 100 ways of arranging the digits so that the 3s are separated.