Hi, hope you can help me. This question has to do with Permutations and Combinations?

Question

A binary number consist of a string of digits that are either 0s o 1s

1) If a string of binary codes is 5 digits long, how many binary numbers are possible if the first digit is a 1?
2)How many different binary numbers can be represented by a string of binary code with 5 digits if the first is 1 and the last two digits are 0?

Answer 1

1) 2^4 =16 You have one fixed digit and 4 digits that can have one of two values. So the answer is 2^4.

2) 2^2 =4 Same reasoning but you have 3 fixed digits and 2 that can be 0 or 1. (00, 01, 10, 11)

In general the answer is 2^n where n is the number of variable digits and n+fixed digits = total digits

Answer 2

Each binary digit doubles the amount of combinations possible.

If there are five positions, but you cannot use one of them, only four positions are really available.

One digit: two possibilities (0 and 1)
Two digits: double that: 4
Three digits: double again: 8
Four digits: 16

Write them down if you doubt:

0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111

Prefix these numbers with a 1, as stated in the query.


Answer for (2): only 2 digits are allowed to change. In the table above, you'll find this means four combinations.
These are:

10000
10100
11000
11100

Same table with some positions x-ed out (may not change):

x00xx
x01xx
x10xx
x11xx


For any number of digits: 2 to the power n (where n is number of digits).

For other numeric systems, like decimal: m to the power n, where m is the amount of different symbols. 10 to the power of 2 (100) is the amount of different combinations for two positions in the decimal system.

2 to the power of 2 (4) is the amount of different combinations for two positions in the binary system.

Answer 3

1) So you know that there are 2 possibilities for each digit of the number (0 or 1). You also know that 1 is the first digit.

The total possible would be: 1*2*2*2*2 = 16

This is because for the first digit you are locked with one possibility (1) while the other four you are limited to 2 (0 and 1).


2) This is a similar question, you just need to figure out which digits are 'locked.'

First - locked (1)
Second -1 and 0 are possible
Third - 1 and 0 are possible
Fourth - locked (0)
Fifth - locked (0)

Therefore: 1*2*2*1*1 = 4

Answer 4

1. Wouldn't that be a 4 digit binary number?

2, Isn't that a 2 digit binary number?

Answer 5

It is arrangements with repetition allowed.

1) 1???? AR(2,4) = 2^4

2) 1??00 AR(2,2) = 2^2