probability question2

explorer

2012-02-15 9:05 pm

Problem 2:
A base ten number is a string of five digits, where the digits are from the
set {1.... 9} but the rst digit cannot be 0 (so 52375 is a valid number
but 02323 and 2323 are not).
(a) How many five-digit base ten numbers are there?
(b) How many five-digit numbers have no two consecutive digits equal?
(c) How many have at least one pair of consecutive digits equal?

回答 (1)

用戶2053

2012-02-17 1:51 am

✔ 最佳答案

a)

9 possible numbers for each digits :9 * 9 * 9 * 9 * 9 = 9⁵ = 59049 numbers.
b)Case 1 :
For no repeat digits ,
9P5 = 15120 numbers.
Case 2 :
1 pair repeat digits (AA) (9 ways) and have no two consecutive digits equal : ( ) B ( ) C ( ) D ( ) , choosing 2 ( ) fill A in them : 4C2 ways , B,C,D : 8P3 ways ,9 * 4C2 * 8P3 = 9 * 6 * 336 = 18144 numbers in this case.
Case 3 :
3 repeat digits(AAA) (9 ways) or 3 repeat digits plus a pair repeat digits ,
and have no two consecutive digits are equal : A ( ) A ( ) A 8 possible numbers for each ( ) .9 * 8 * 8 = 576 numbers in this case.
Total : 15120 + 18144 + 576 = 33840 numbers.
c)59049 - 33840 = 25209 numbers.

👍 0 👎 0