Descending order means from highest to lowest.

$$\therefore$$ Descending order of digits 1, 2, 3, 5, 7 is = 7, 5, 3, 2, 1.

(1) Number starts with 7 :

JEE Main 2023 (Online) 29th January Morning Shift Mathematics - Permutations and Combinations Question 95 English Explanation 1

$$\therefore$$ Total possible numbers start's with 7 are = ⁵C₁ $$\times$$ ⁵C₁ $$\times$$ ⁵C₁ $$\times$$ ⁵C₁ = 5⁴

(2) Number of starts with 5 :

JEE Main 2023 (Online) 29th January Morning Shift Mathematics - Permutations and Combinations Question 95 English Explanation 2

$$\therefore$$ Total possible numbers start's with 5 are = ⁵C₁ $$\times$$ ⁵C₁ $$\times$$ ⁵C₁ $$\times$$ ⁵C₁ = 5⁴

(3) Number starts with 37 :

JEE Main 2023 (Online) 29th January Morning Shift Mathematics - Permutations and Combinations Question 95 English Explanation 3

$$\therefore$$ Total possible numbers starts with 3 and 7 are = ⁵C₁ $$\times$$ ⁵C₁ $$\times$$ ⁵C₁ = 5³

(4) Number starts with 357 :

JEE Main 2023 (Online) 29th January Morning Shift Mathematics - Permutations and Combinations Question 95 English Explanation 4

$$\therefore$$ Total possible numbers starts with 357 are = ⁵C₁ $$\times$$ ⁵C₁ = 5²

(5) Number starts with 355 :

JEE Main 2023 (Online) 29th January Morning Shift Mathematics - Permutations and Combinations Question 95 English Explanation 5

$$\therefore$$ Total possible numbers starts with 355 are = ⁵C₁ $$\times$$ ⁵C₁ = 5²

(6) Number starts with 3537 :

JEE Main 2023 (Online) 29th January Morning Shift Mathematics - Permutations and Combinations Question 95 English Explanation 6

$$\therefore$$ Total possible numbers starts with 3537 are = ⁵C₁ = 5

(7) Number starts with 3535 :

JEE Main 2023 (Online) 29th January Morning Shift Mathematics - Permutations and Combinations Question 95 English Explanation 7

$$\therefore$$ Total possible numbers starts with 3535 are = ⁵C₁ = 5

(8) Next number is 35337 which is the required number.

$$\therefore$$ Position of the number 35337 is = 5⁴ + 5⁴ + 5³ + 5² + 5² + 5 + 5 + 1 = 1436

Permutations And Combinations

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