How many integers are there from 499 to 999 b How many odd i

How many integers are there from 499 to 999? (b) How many odd integers are there from 100 to 999? (c) How many integers from 100 to 999 have distinct digits? (d) How many odd integers from 100 to 999 have distinct digits? (e) What is the probability that a randomly chosen three-digit integer has distinct digits? You may leave your answer as a fraction. (f) What is the probability that a randomly chosen three-digit integer has distinct digits and is odd? You may leave your answer as a fraction.

Solution

a) No. of integers from 499 to 999 = 999 - 499 + 1 = 501

b) Odd integers from 100 to 999 are 101, 103,.....,999

a = 101, d = 2 and a_n = 999

So by Arithmetic progression, a_n = a + (n - 1)d = 101 + 2(n - 1)

=> 999 - 101 = 2(n - 1)

=> n = 450

c) At hundredth place we have 9 choices of placing a no. because 0 can\'t come. On tenth place we have 9 choices again as we can\'t place the number that we have placed at 100th place. Similarly we have 8 choices for unit place.

So no. of distinct digit numbers from 100 to 999 = 9*9*8 = 648

d) At unit place we have 5 choices (1,3,5,7,9). Now at hundredth place we have 8 choices of placing a no. because 0 can\'t come and one of the remaining is occupied at unit place. On tenth place we have 8 choices again as we can\'t place the number that we have placed at 100th place and unit place.

So no. = 8*8*5 = 320

e) 0 at the will not be considered as 3 digit integer.

At hundredth place we have 9 choices of placing a no. because 0 can\'t come. On tenth place we have 9 choices again as we can\'t place the number that we have placed at 100th place. Similarly we have 8 choices for unit place.

So probability = 9/10 * 9/10 * 8/10 = .648

f) 0 at the will not be considered as 3 digit integer.

At unit place we have 5 choices (1,3,5,7,9). Now at hundredth place we have 8 choices of placing a no. because 0 can\'t come and one of the remaining is occupied at unit place. On tenth place we have 8 choices again as we can\'t place the number that we have placed at 100th place and unit place.

So probability = 8/10 * 8/10 * 5/10 = .320