Permutations & CombinationsmediumFree

Nine-digit numbers and the ratio n1/n2 for the digits at two places | JEE Advanced

JEE Maths question with a full step-by-step solution.

Question
Let n=x1x2x3…x9n = x_1x_2x_3\ldots x_9 be formed using the digits 1,2,…,91,2,\ldots,9, each digit being used once only. If n1n_1 is the number of possible nn for which 1<∣x3−x4∣<51<\left|x_3-x_4\right|<5 and n2n_2 is the number of possible nn for which ∣x3−x4∣=1\left|x_3-x_4\right| = 1, then [n1n2]\left[\dfrac{n_1}{n_2}\right] is equal to (where [ ⋅ ][\,\cdot\,] is G.I.F.
Solution
Answer: 2
Step 1: Only x3x_3 and x4x_4 are constrained. Once an ordered pair of distinct digits is placed there, the remaining seven digits fill the remaining seven positions in 7!7! ways, the same factor in both counts, so it cancels in the ratio. Step 2: For n2n_2, ∣x3−x4∣=1\left|x_3-x_4\right| = 1. The unordered pairs of consecutive digits are {1,2},{2,3},…,{8,9}\{1,2\},\{2,3\},\ldots,\{8,9\}, eight of them, and each can be placed in 22 orders:
n2=2×8×7!=16⋅7!n_2 = 2\times8\times7! = 16\cdot7!
 Step 3: For n1n_1, the inequality 1<∣x3−x4∣<51<\left|x_3-x_4\right|<5 is strict at both ends, so the difference is 22, 33 or 44 and nothing else. For a difference dd the unordered pairs are {k,k+d}\{k,k+d\} with 1≤k≤9−d1 \le k \le 9-d, so there are 9−d9-d of them:
d=2: 7,d=3: 6,d=4: 5d=2:\ 7 ,\qquad d=3:\ 6 ,\qquad d=4:\ 5
 Doubling for order,
n1=2(7+6+5)×7!=36⋅7!n_1 = 2\left(7+6+5\right)\times7! = 36\cdot7!
 Step 4: The 7!7! cancels.
n1n2=3616=94=2.25\frac{n_1}{n_2} = \frac{36}{16} = \frac94 = 2.25
[2.25]=2\left[2.25\right] = 2
 (Running over all 9×8=729\times8 = 72 ordered pairs (x3,x4)\left(x_3,x_4\right) of distinct digits gives 3636 and 1616 directly.) Answer: 22
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Permutations & Combinations · medium
The number of 44-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is
Permutations & Combinations · medium
The number of seven-digit numbers that can be formed by using the digits 1,2,3,51, 2, 3, 5 and 77 such that each digit is used at least once, is
Permutations & Combinations · medium
The number of ways of forming a queue of 44 boys and 33 girls such that all the girls are not together, is
Permutations & Combinations · medium
Two players AA and BB play a series of games of badminton. The player who wins 55 games first wins the series. Assuming no game ends in a draw, the number of ways in which player AA wins the series is

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath