Question

Q1. City K's home phone numbers have 6 digits. In a home phone number, each digit can be any number of 0,1,..., 9, except that a phone number many not start with the following sequences: a) reserved for emegency services: 110, 119, 120, 120. b) reserved for domestic and international dial prefixes: 0. At most how many distinct home phones can this system accommoodate? For Example 120193 and 018483 are invalid.

Q2. In how many ways can we assign n identical balls into k distinct boxes. (n>=k), leaving no empty boxes?

Q3. How many children should a family plan to have so that the probability of having at least one child of each sex is at least 0.95?( Assume that both sexes are equally likely, and the sex of each child does not depend on the other children?

Answer 1

Q1.

There are in total 10 digits. The number of phone numbers made by 6 digits would be . Now we will reduce the number of phone numbers with 110, 119 and 120 in beginning. For each combination the number of phone numbers would be . To reduce the phone numbers starting with 0 we will reduce numbers. Hence the total number of valid phone numbers would be given by,

Q2.

First we will select k balls out of n balls. The number of ways to select this would be C(N,k). Since each box is distinct there will be another factor of k! for rearrangement. Hence the number of ways of assigning n identical balls into k distinct boxes would b given by,

Q3.

Let us assume that there are N children in family. To find the probability that there are atleast one child of each sex in the family , we will simply substract the probability that all children are boys/girls. We have,