7. (15 pts) (a) How many odd integers are there from 1 through 99? (b) How many

Question

7. (15 pts) (a) How many odd integers are there from 1 through 99? (b) How many integers from 1 through 99 have distinct digits? c) How ?nany odd integers from 1 through 99 have distinct digits? 8. (8 pts) How many integers from 1 through 100 are multiples of 2 or multiples of 5? 9. (11 pts) Suppose the group of ten people consists of four men and six women (a) How many four-person teams contain at least one man? ) Suppose two members of the group of twelve insist on working as a pair-any team must contain either both or neither. How many four-person teams can be formed?

Explanation / Answer

7.

(a)

1,3,5,7..

Odd and even number occur consecutively one after another.

If there are 2n consecutive numbers, n out of them would have to even and n would be odd.

So from 1 though 98, there are 49 odd integers

and since 99 is odd too, in total we have 50 odd integers.

(b)

Number of integers with distinct digits = total number of integers - total number of inegers with same digits

total number of inegers with same digits = {11,22,33...99} = 9 numbers

Number of integers with distinct digits = 99 - 9 = 90

(c)

Integers which are odd and have distinct digits = Odd integer - Odd integers with same digits

Odd integers with same digits = 11,33,55,77,99 = {5}

Integers which are odd and have distinct digits = 50 - 5 = 45

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