Let S denote the set of four-digit numbers made from the digits 0, 2, 3, 5, and 6, where the first digit cannot be zero.
a. How many numbers are there in S?
b. How many numbers in S have no repeated digit?
c. How many of those in part (b) are even?
d. How many of those in part (b) are greater than 4000?
Answers
Answer:
(a) option is correct so, you have go in this option
Given : S denote the set of four-digit numbers made from the digits 0, 2, 3, 5, and 6, where the first digit cannot be zero.
To Find : a. How many numbers are there in S?
b. How many numbers in S have no repeated digit?
c. How many of those in part (b) are even?
d. How many of those in part (b) are greater than 4000?
Solution:
0, 2, 3, 5, and 6
5 digits
Repeated digits
1st digit can be in 4 ways
all other digits can be in 5 ways
4 x 5 x 5 x 5 = 500
500 numbers are there in S
No repetition
1st digit can be in 4 ways
3 digits out of 4 can be selected in ⁴C₃ = 4 ways
remaining 3 digits can be in 3!
= 4 * 4 * 3! = 96 numbers
96 numbers in S have no repeated digit
Even number in part b
ending with 0 - 1st 3 numbers can be in 4! = 24
ending with 2 - 1st number in 3 ways and remaining 3! = 18
ending with 6 - 1st number in 3 ways and remaining 3! = 18
18 + 18 + 24 = 60
60 numbers are even when digits are not repeated
greater than 4000
starting with 5 or 6 - 2 ways
remaining 3 digits in 4 ! = 24 ways
2 * 24 = 48 numbers greater than 4000
Learn More:
how many different 6 digit numbers can be formed using digits in the ...
brainly.in/question/13738553
A two digit number is formed with digits 2, 3, 5, 7, 9 without repetition ...
brainly.in/question/4334342