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
Or would it be:
3 for the first number (either a 3,4 or 5)
10 for the second number (a 0 to 9)
10 for the third number (a 0 to 9)
10 for the fourth number (a 0 to 9)
take off numbers that end in 000 so 3000, 4000, 5000 (so take off 3)
=
3 x 10 x 10 x 10 -3
3000-3 = 2997
3*10*10*9
check that. The last nine allows no 000
To find the number of different sequences of 4 digits, where the first digit must be 3, 4, or 5 and the sequence may not end in 000, we can break down the problem into smaller steps.
Step 1: Determine the choices for the first digit.
Since the first digit must be 3, 4, or 5, there are 3 choices for this digit.
Step 2: Determine the choices for the second, third, and fourth digits.
For each of the second, third, and fourth digits, we have 10 choices, ranging from 0 to 9. Since repetition of digits is allowed, each digit has 10 choices.
Step 3: Remove the invalid choice of ending in 000.
Out of the total possible sequences, we need to remove the ones that end in 000. Since repetition is allowed, we have already included sequences like 3000, 4000, and 5000 as valid choices. However, we need to remove the specific sequence 0000 as it violates the condition. So we subtract 1 from the total count.
Step 4: Calculate the total number of sequences.
To calculate the total number of sequences, we multiply the choices from each step:
Total = Choices for the first digit * Choices for the second digit * Choices for the third digit * Choices for the fourth digit - Invalid choice for ending in 000
Total = 3 * 10 * 10 * 10 - 1
Total = 3000 - 1
Total = 2999
Therefore, there are 2999 different sequences of 4 digits that meet the given conditions.
To find the number of different sequences of 4 digits satisfying the given conditions, we can break down the problem into two parts:
1. Finding the number of choices for the first digit: Since the first digit must be 3, 4, or 5, there are three possible choices.
2. Finding the number of choices for the remaining three digits: Since repetition of digits is allowed, each of the remaining three digits can be selected from the numbers 0 to 9, meaning there are 10 choices for each digit.
However, we need to exclude the case where the sequence ends in 000. To calculate this, we can subtract the number of sequences that end in 000 from the total number of sequences.
To find the number of sequences that end in 000, we need to consider the choices for the last three digits. Since we have excluded the digit 0 as a choice for the first digit, there are only nine choices for each of the remaining three digits (1 to 9). Therefore, the number of sequences that end in 000 is 9*9*9 = 729.
Now, let's calculate the total number of sequences satisfying the given conditions:
Total number of sequences = Number of choices for the first digit * Number of choices for the remaining three digits - Number of sequences that end in 000
= 3 * 10 * 10 * 10 - 729
= 3000 - 729
= 2271
Therefore, there are 2271 different sequences of 4 digits that satisfy the given conditions.