how many different sequences of 4 digits possible if the first digit must be 3,4 or 5 and if the sequence may not end in 000? repetition of digits is allowed.

number of sequences starting with 3,4, or 5 without any restrictions at the end

= 3x10x10x10 = 3000
but we don't want the numbers 3000, 4000 and 5000

so the number of sequences as stated is
3000 - 3 = 2997

To determine the number of different sequences of 4 digits meeting the given conditions, we can break down the problem into steps:

Step 1: Count the possibilities for the first digit.
Since the first digit must be either 3, 4, or 5, there are 3 possible choices.

Step 2: Count the possibilities for the second, third, and fourth digits.
Since repetition of digits is allowed, there are 10 possibilities for each of the three remaining digits (0-9).

Step 3: Exclude the possibility of the sequence ending in 000.
If the sequence may not end in 000, we need to subtract the single invalid option from our total count.

Step 4: Calculate the total number of sequences.
Multiply the counts from steps 1, 2, and 3 together to get the total number of sequences.

Step 1: 3 possibilities (3, 4, or 5) for the first digit.
Step 2: 10 possibilities for each of the three remaining digits (repetition allowed).
Step 3: 1 invalid possibility (000) to exclude.
Step 4: Multiply the counts together: 3 * 10 * 10 * 10 - 1 = 2,999 possible sequences.

Therefore, there are 2,999 different sequences of 4 digits possible, where the first digit is either 3, 4, or 5, and the sequence does not end in 000.