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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME II, 2015 based on Sequence and permutations.

Call a permutation \(a_1,a_2,....,a_n\) of the integers 1,2,...,n quasi increasing if \(a_k \leq a_{k+1} +2\) for each \(1 \leq k \leq n-1\), find the number of quasi increasing permutations of the integers 1,2,....,7.

- is 107
- is 486
- is 840
- cannot be determined from the given information

Sequence

Permutations

Integers

But try the problem first...

Answer: is 486.

Source

Suggested Reading

AIME II, 2015, Question 10

Elementary Number Theory by David Burton

First hint

While inserting n into a string with n-1 integers, integer n has 3 spots where it can be placed before n-1, before n-2, and at the end

Second Hint

Number of permutations with n elements is three times the number of permutations with n-1 elements

or, number of permutations for n elements=3 \(\times\) number of permutations of (n-1) elements

or, number of permutations for n elements=\(3^{2}\) number of permutations of (n-2) elements

......

or, number of permutations for n elements=\(3^{n-2}\) number of permutations of {n-(n-2)} elements

or, number of permutations for n elements=2 \(\times\) \(3^{n-2}\)

forming recurrence relation as the number of permutations =2 \(\times\) \(3^{n-2}\)

for n=3 all six permutations taken and go up 18, 54, 162, 486

Final Step

for n=7, here \(2 \times 3^{5} =486.\)

as

sds

- https://www.cheenta.com/inequations-and-conditions-isi-b-stat-tomato-problem/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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