P4: In the following problem, numbers do not begin with zero. (ex: 123456 is a 6-digit number but 012345
is not.) You may leave your answers unsimplified.
(a)
How many 6-digit numbers are there?
(b)
How many 6-digit even numbers are there?
(c)
How many 6-digit numbers have all odd digits?
(d)
How many 6-digit numbers have all even digits?
(e)
How many 6-digit numbers have all distinct digits?
(f)
The numbers 3514153 and 8068608 are called
palindromic
because they read the same forwards and
backwards.
i.
How many 7-digit palindromic numbers are there?
ii.
How many 7-digit even palindromic numbers are there?
a) Number of 6 digit numbers = 9x105
= 900,000
b) Number of 6 digit even numbers = 9x104x5
= 450,000
c) Number of 6 digit numbers will all odd digits = 56
= 15,625
d) Number of 6 digit numbers will all even digits = 4 x 55 (first digit cannot be 0)
= 12,500
e) Number of 6 digit numbers with all distinct digits = 9 x 9P5
= 9 x 15120
= 136,080
f) In palindrome of 7 digits, the first 4 digits shall have choices and the last 3 will not have choice as it is same as the first 3 which is already selected
i. Number of seven digit palindromes = 9x10x10x10x1x1x1
= 9,000
ii. Number of seven digit even lindromic numbers = 4 x 10x10x10 x 1x1x1
= 4,000
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