Suppose you made a list of all the possible six-digit numbers that use only the

Question

Suppose you made a list of all the possible six-digit numbers that use only the digits 1,2,3,4,5, and 6. For example, 362214, 153653, 444365, 156324, 222222 are just a few of the numbers on this list. Answer the following questions about your hypothetical list.

a) How many total possible numbers on the list are there?

b) How many possible numbers without any repeated digits are there?

c) How many possible numbers with digits that repeat are there?

d) How many possible numbers on the list are multiples of 5?

e) What is the probebility that a number chosen randomly from the list will not be a multiple of 5?

Explanation / Answer

(a) Total possible numbers (with repetition) = 66.

The first place can take any 5 values, the second place any 5, third place any five.... = 6 x 6 x 6 x 6 x 6 x 6 = 66 = 46,656.

(Also we can use the rule, that number of ways of arranging n distinct numbers all used together with repetition = nn). Here 6 numbers so 66 = 46,656.

(b) No digits repeated. All are unique. The first place can take any 6, second place any 5, third place any 4...

6 x 5 x 4 x 3 x 2 x 1 = 720 (Also we can use the rule, that number of ways of arranging n distinct numbers all used together without repetition= n!). Here 6 numbers so 6! = 720.

(c) All we need to do is subtract the total numbers and numbers formed where there is no repetition.

= 46656 - 720 = 45,936

(d) Here the last digit has to be 5. Therefore the other 5 position can take any of the 6 digits ( as we are having repetition) = 6 x 6 x 6 x 6 x 6 = 65 = 7,776.

(e) To get a number which will not be a multiple of 5. Subtract all multiples of 5 from the total possible numbers = 46,656 - 7,776 = 38,880.

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