The number of integers between 100 and 1000 s | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the three-digit number be $N = 100a + 10b + c$,where $a \in \{1, 2, \dots, 9\}$ and $b, c \in \{0, 1, \dots, 9\}$.We need to find the number o

Vedclass · Accepted Answer

$70$

Vedclass · Answer

(D) Let the three-digit number be $N = 100a + 10b + c$,where $a \in \{1, 2, \dots, 9\}$ and $b, c \in \{0, 1, \dots, 9\}$.We need to find the number of solutions to $a + b + c = 14$ under the given constraints.Using the stars and bars method with inclusion-exclusion:Let $a' = a - 1$,so $a' \geq 0$. Then $(a' + 1) + b + c = 14 \implies a' + b + c = 13$,where $0 \leq a' \leq 8$,$0 \leq b \leq 9$,and $0 \leq c \leq 9$.The total number of non-negative integer solutions to $a' + b + c = 13$ is $\binom{13+3-1}{3-1} = \binom{15}{2} = 105$.Now,subtract cases where variables exceed their limits:Case $1$: $a' \geq 9$. Let $a'' = a' - 9$,then $a'' + b + c = 4$. Number of solutions = $\binom{4+3-1}{2} = \binom{6}{2} = 15$.Case $2$: $b \geq 10$. Let $b' = b - 10$,then $a' + b' + c = 3$. Number of solutions = $\binom{3+3-1}{2} = \binom{5}{2} = 10$.Case $3$: $c \geq 10$. Let $c' = c - 10$,then $a' + b + c' = 3$. Number of solutions = $\binom{3+3-1}{2} = \binom{5}{2} = 10$.Total valid solutions = $105 - (15 + 10 + 10) = 105 - 35 = 70$.

The number of integers between $100$ and $1000$ such that the sum of their digits is equal to $14$ is:

Similar Questions

If $^{n^2 - n}C_2 = ^{n^2 - n}C_{10}$,then $n = $

The number of four-digit numbers that can be formed using all the digits from $1$ to $9$ (excluding zero) such that every number has exactly $2$ distinct digits is:

The number of ordered triplets of positive integers which are solutions of the equation $x + y + z = 100$ is

If $x, y$ and $r$ are positive integers,then $^x{C_r} + ^x{C_{r-1}} ^y{C_1} + ^x{C_{r-2}} ^y{C_2} + \dots + ^y{C_r} = $

If $10 \cdot ^nC_2 = 3 \cdot ^{n+1}C_3$,then the value of $n$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module