# Since when is 5^7 =8125?

if a number is a multple of ten, that means the last digit is zero. The leading digit cannot be zero.

9*10^5*1

on c, again, the leading digit cannot be zero.

d is right

e. no leading zero is allowed. 9*9*8*7*6*5*4= ?

can someone correct this for me parts A-E and if i am wrong can you help me....please

How many 7-digit telephone numbers are possible if the first digit cannot be zero and.....

a) only odd digits may be used?

5^7= 8125

my answer: 8125

b) the telephone number must be multiple of 10(that is , it must end in zero)?

10^6*1=1,000,000

my answer: 1,000,000

c) the telephone number must be a multiple of 100?

10^5*1*1=100,000

my answer 100,000

d) the first 3 digits are 481?

1*1*!*10^4=10,000

my answer 10,000

e) no repetitions are allowd?

10 combination 7 = 120

my answer: 120

oops for the part a: 5^7 = 78125

part e: 544320

i need help with c

## Let's go through each part and correct any mistakes:

a) For this part, you are correct. The number of 7-digit telephone numbers possible using only odd digits is indeed 5^7, which is equal to 78125.

b) For this part, you made a mistake. The number of 7-digit telephone numbers that must end in zero (i.e., multiples of 10) is 10^6, not 10^6 * 1. So the correct answer is 10^6, which is equal to 1,000,000.

c) For this part, it seems you made an error in your calculation. If a telephone number must be a multiple of 100, it means it must end in two zeros. So you have one required digit at the end (zero) and you can choose any digit (0-9) for the leading digit. That means you have 10 choices for the leading digit and 1 choice for the last digit, giving a total of 10 * 1 = 10 possible telephone numbers.

d) For this part, your calculation is correct. Since the first 3 digits must be 481, you have 1 choice for each of those digits and then you have 10 choices for each of the remaining 4 digits. So the total number of telephone numbers is 1 * 1 * 1 * 10^4 = 10,000.

e) For this part, your calculation is incorrect. The question states that no repetitions are allowed, which means you cannot use the same digit twice. Since the first digit cannot be zero, you have 9 choices for the first digit, then 9 choices for the second digit (excluding the one you used for the first digit), 8 choices for the third digit, and so on. Therefore, the total number of telephone numbers is 9 * 9 * 8 * 7 * 6 * 5 * 4 = 326,592.

To summarize:

a) 78125

b) 1,000,000

c) 10

d) 10,000

e) 326,592

## Let's go through each part and correct any errors:

a) To find the number of possible 7-digit telephone numbers if only odd digits may be used, we need to consider that there are 5 odd digits (1, 3, 5, 7, 9) and 7 positions to fill. Since we can repeat the odd digits, we need to use the concept of combinations with repetition. The formula for combinations with repetition is (n + r - 1) choose r, where n is the number of options and r is the number of positions to fill.

So for part a, it should be 5^7 = 5 * 5 * 5 * 5 * 5 * 5 * 5 = 78125.

b) For a telephone number to be a multiple of 10, it needs to end in zero. Since the last digit must be zero, we have only 1 option for the last digit (0). For the remaining 6 digits, each digit can be chosen freely (0-9). Thus, the number of possible 7-digit telephone numbers that end in zero is 10^6 = 1,000,000.

c) To find the number of possible telephone numbers that must be a multiple of 100, we need to consider that the last two digits must be zeros. So for the last two digits, we have only 1 option (00). For the remaining 5 digits, each digit can be chosen freely (0-9). Thus, the number of possible telephone numbers that are a multiple of 100 is 10^5 = 100,000.

d) When the first 3 digits are fixed as 481, we have only 1 option for each of the first 3 digits. For the remaining 4 digits, each digit can be chosen freely (0-9). Thus, the number of possible telephone numbers with the first 3 digits as 481 is 1 * 1 * 1 * 10^4 = 10,000.

e) To find the number of possible telephone numbers with no repetitions allowed, we need to consider that each digit must be unique. For the first digit, we have 9 options (1-9) since zero cannot be the leading digit. For the second digit, we have 9 options (0-9 excluding the digit used for the first digit). For the third digit, we have 8 options (0-9 excluding the two digits used for the first two digits). This pattern continues until the seventh digit, where we have 3 options (0-9 excluding the six digits used before). Thus, the number of possible telephone numbers with no repetitions allowed is 9 * 9 * 8 * 7 * 6 * 5 * 4 = 544320.

So the corrected answers for the parts are:

a) Corrected: 78125

b) Correct: 1,000,000

c) Corrected: 100,000

d) Correct: 10,000

e) Corrected: 544320