In each of the following cases, determine the number of four-digit integers that

Question

In each of the following cases, determine the number of four-digit integers that satisfy the given condition. A.) Unrestricted B.) The digits are all even. C.) There are no repeated digits D.) No two consecutive digits are equal E.) The integer is even and no two consecutive digits are equal. In each of the following cases, determine the number of four-digit integers that satisfy the given condition. A.) Unrestricted B.) The digits are all even. C.) There are no repeated digits D.) No two consecutive digits are equal E.) The integer is even and no two consecutive digits are equal. In each of the following cases, determine the number of four-digit integers that satisfy the given condition. A.) Unrestricted B.) The digits are all even. C.) There are no repeated digits D.) No two consecutive digits are equal E.) The integer is even and no two consecutive digits are equal.

Explanation / Answer

A)

Each digit has 10 possibilities except the first which cannot be 0 else we wont ahve 4 digit integer

SO, 10^3*9=9000 integers

B)

There are 5 even digits

So 5 possibilities for each digit except the first which cannot be 0

So, 4*5^3=4*125=500 integers

C)

First can have 9 digits (Cannot be 0)

Then 9 left to chose from for second

And then 8 for enxt

And 7 for last

So, 9*9*8*7=4536

D)

First digit can have 9 possibilities.

Second can be one of the 9 digits other than first digit

Next can be 9 digits other than second

Next can be 9 digits other than third

So, 9^4=6561

E)

So digit ends with an even digits so 5 possibilties for last digit

Third can be one of 5 odd digits or 4 remaining even digits =digits

Second can be any of the 9 digits other than third

1st can be any of the 9 digits other than third

So,9^3*5=3645 integres

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