Provide an appropriate response.
-How many different sequences of 4 digits are 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.
A) 5000
B) 2999
C) 1512
D) 2997
E) 2000

Question

Quizplus · Accepted Answer

There are 3 choices for the first digit, and 10 choices for each of the remaining three digits, so there are 3 × 10 × 10 × 10 = 3000 possible sequences. However, we must subtract the number of sequences that end in 000, which is 3 × 10 × 10 = 300. Therefore, the total number of possible sequences is 3000 − 300 = 2700. There are three options for the first digit (3, 4, or 5) and 10 options for each of the remaining three digits (since repetition is allowed). This gives us a total of 3×10×10×10=3000 possible sequences. However, we must subtract the number of sequences that end in 000 (which is just the number of sequences of three digits, since the first digit must be fixed). There are 3×10×10=300 such sequences. Thus, the total number of different sequences of 4 digits that are possible is 3000−300=2700.

Provide an Appropriate Response