QJEEAdv22Made28: JEE Advanced 2022 Mathematics - Class 11 JEEAdv22Made28 Solution: The number of 4-digit integers in the closed interval [2022,…

This is a 4-digit integer whose 0th digit called D, 10th digit called C, 100th digit called B and 1000th digit called A.

Given range for possible number is 2022 to 4482.

So, position A can have digits 2, 3 or 4.

0, 6 and 7 can't put in A as number starts with 0, 6 or 7 don't fall in the range between 2022 to 4482.

So, Position A can be filled with 2, 3 or 4 in $${}^3{C}_1=3$$ ways.

Position B can be filled by one of 0, 2, 3, 4, 6, 7

So, total possible ways for position B = $${}^6{C}_1=6$$ ways

Similarly, Position C can be filled in $${}^6{C}_1=6$$ ways

Position D can be filled in $${}^6{C}_1=6$$ ways

$$\therefore$$ Total possible integer numbers starting with 2 or 3 or 4

$$=3\times 6\times 6\times 6=3\times 216=648$$

Now lets find those numbers which starts with 2 and 4 but don't fall in the range between 2022 to 4482.

Situation 1 :

$$\therefore$$ Total possible numbers starting with 200 are = 6

Situation 2 :

$$\therefore$$ Total possible number in this case = 1

Situation 3 :

$$\therefore$$ Total numbers in this case $$=2\times 6\times 6=72$$

$$\therefore$$ Total number that don't fall in the range 2022 to 4482 which starts with 2 or 4 are

$$=6+1+72=79$$

$$\therefore$$ Total numbers falls in the range 2022 to 4482

$$=648-79$$

$$=569$$