Answer:
(a) 1000
(b) 4500
(c) 4536
(d) 3000
(e) 2829
(f) 6171
(g) 1543
(h) 257
Step-by-step explanation:
Let X be positive integers between 1000 and 9999 which contains 9000 integers.
(a)The integers divided by the number of elements 9 is:
Use the quotient rule = absolute number of integers(X)/ number of elements
i = |9000|/9 = 1000
(b) Similar we use the quotient rule but we use an even number since even numbers are divided by 2 we can use 2.
i = |9000|/2 = 4500
(c) There are 10 digits
The first digit cannot be be zero so you can divide is 9 ways
The second digit can be zero but not the same as the first digit and therefore 9 ways
The third digit 8 ways because the second digit cannot be the same as the first and second.
The fourth digit 7 ways for the same reason as the third digit.
Using the product rule = i = 9*9*8*7 = 4536 have distinct digits
(d) Use the quotient rule = i = 9000/3 = 3000
(e) Use the quotient rule = i5 = 9000/5 = 1800
Use the quotient rule = i7 = 9000/7 = 1286
the integers divisible by 5 and 7 is 5*7 = 35
i35 = 9000/35 = 257
Number of integers divisible by 5 or 7 can be determined by the subtraction rule therefore:
i = i5+i7-i35 = 1800+1286-257 = 2829
(f) The integers divisible not divided 5 or 7 is the same as the integers not divisible by 3 or 4:
i(not 5 or 7) = |X| - i(5 or 7) = 9000-2829 = 6171
(g) Integers divisible by 11 but not 7 are the same as integers divisible by 11 but not divisible by 7 and 11.
i(5 not by 7) = i5-i35 = 1800 - 257 = 1543
(h) i(5 and 7) = i35 = 257