Let P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set $P$ is: (1) 173 (2) 164 (3) 158 (4) 161
Q81. The number of 3-digit numbers, formed using the digits $2,3,4,5$ and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to $\_\_\_\_$
Q1 Using the five numerals $0,1,2,3,4$, we are to make four-digit integers. (Note that "0123", etc. are not allowed.) (1) The total possible number of integers where the digits are all different numerals is $\mathbf{AB}$. Among them, the total number of integers that do not use 0 is $\mathbf{CD}$. (2) If we are allowed to use the same numeral repeatedly, then the total possible number of four-digit integers is $\mathbf{EFG}$. Among them (i) the total number of integers that use both 1 and 3 twice is $\mathbf{H}$, (ii) the total number of integers that use both 0 and 4 twice is $\mathbf{I}$, (iii) the total number of integers that use both of two numerals twice is $\mathbf{JK}$.
We define the digit sum of a non-negative integer to be the sum of its digits. For example, the digit sum of 123 is $1 + 2 + 3 = 6$. (i) How many positive integers less than 100 have digit sum equal to 8 ? Let $n$ be a positive integer with $n < 10$. (ii) How many positive integers less than 100 have digit sum equal to $n$ ? (iii) How many positive integers less than 1000 have digit sum equal to $n$ ? (iv) How many positive integers between 500 and 999 have digit sum equal to 8 ? (v) How many positive integers less than 1000 have digit sum equal to 8 , and one digit at least 5 ? (vi) What is the total of the digit sums of the integers from 0 to 999 inclusive?
4. License plates in a certain region consist of six characters: the first two are uppercase English letters other than O, and the last four are Arabic numerals from 0 to 9. However, three consecutive 4's are not allowed. For example, AA1234 and AB4434 are valid license plates, while AO1234 and AB3444 are not. The number of license plates with first character A and last character 4 is (1) $25 \times 9 ^ { 3 }$ (2) $25 \times 9 ^ { 2 } \times 10$ (3) $25 \times 900$ (4) $25 \times 990$ (5) $25 \times 999$
A department store holds a Father's Day card drawing promotion with the following rules: The organizer prepares ten cards numbered $1, 2, \ldots, 9$, of which there are two cards numbered 8, and only one card for each other number. Four cards are randomly drawn from these ten cards without replacement and arranged from left to right in order to form a four-digit number. If the four-digit number satisfies any one of the following conditions, a prize is won: (1) The four-digit number is greater than 6400 (2) The four-digit number contains two digits 8 For example: If the four cards drawn are numbered $5, 8, 2, 8$ in order, then the four-digit number is 5828, and a prize is won. According to the above rules, there are (11-1)(11-2)(11-3)(11-4) four-digit numbers that can win prizes.