Question

1. If you were to toss a fair coin 5 times, how many different possible sequences of flips would there be?

2. Suppose that employees at a large company are assigned an ID number which contains 5 numbers and 2 letters. How many possible combinations are there in this system?

Answer 1

1. For single fair coin toss we have 2 outcomes head or tail i.e. {"H","T"}

Using permutation we can find total possible sequences of flips:

Flip 1 : 2 choices (H or T), Flip 2: 2 choices (H or T), Flip 3: 2 choices (H or T),

Flip 4: 2 choices (H or T) & Flip 5: 2 choices (H or T)

Multiply the number of choices for each coin flip, we get the total possible sequences =  2*2*2*2*2 =

2⁵ = 32.

The 32 possible outcomes are :

{ "H" "H" "H" "H" "H"} {"H" "H" "H" "H" "T"} {"H" "H" "H" "T" "H"} {"H" "H" "H" "T" "T"}

{ "H" "H" "T" "H" "H"} {"H" "H" "T" "H" "T"} { "H" "H" "T" "T" "H"} {"H" "H" "T" "T" "T"}

{"H" "T" "H" "H" "H"} {"H" "T" "H" "H" "T"} {"H" "T" "H" "T" "H"} {"H" "T" "H" "T" "T"}

{ "H" "T" "T" "H" "H"} { "H" "T" "T" "H" "T"} {"H" "T" "T" "T" "H"} {"H" "T" "T" "T" "T"}

{ "T" "H" "H" "H" "H"} {"T" "H" "H" "H" "T"} {"T" "H" "H" "T" "H"} {"T" "H" "H" "T" "T"}

{ "T" "H" "T" "H" "H"} {"T" "H" "T" "H" "T"} {"T" "H" "T" "T" "H"} {"T" "H" "T" "T" "T"}

{ "T" "T" "H" "H" "H"} {"T" "T" "H" "H" "T"} {"T" "T" "H" "T" "H"} {"T" "T" "H" "T" "T"}

{ "T" "T" "T" "H" "H"} {"T" "T" "T" "H" "T"} {"T" "T" "T" "T" "H"} {"T" "T" "T" "T" "T"}.

2. There 26 choices (A to Z) of letters and 10 choices of digits (0,1,2,3,4,5,6,7,8,9).

Since exact format of ID is not given (i.e. letters followed by digits or digits followed by letters etc.) eg. S896D5 or SD8965 or 8965SD etc.

So, there are 7! choices of arranging 2 letters and 5 digits in ID.

Hence, possible combinations in this system = 26*26*10*10*10*10*10*7! = 67600000 * 7! =  67600000 * 5040 = 340704000000.