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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_24 / Let K be the number of sequences A₁, A₂, …, A_(n) such that n is a positive integer less than or equal to 10, each A_(i) is a subset of {1,2,3,…,10}, and A_(i−1) is a subset of …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let

K

be the number of sequences

A_1

A_2

\dots

A_n

such that

n

is a positive integer less than or equal to

10

, each

A_i

is a subset of

\{1, 2, 3, \dots, 10\}

, and

A_{i-1}

is a subset of

A_i

for each

i

between

2

and

n

, inclusive. For example,

\{\}

\{5, 7\}

\{2, 5, 7\}

\{2, 5, 7\}

\{2, 5, 6, 7, 9\}

is one such sequence, with

n = 5

.What is the remainder when

K

is divided by

10

Plain-text mathematical notation (without MathML)

Let K be the number of sequences A₁, A₂, …, A_(n) such that n is a positive integer less than or equal to 10, each A_(i) is a subset of {1,2,3,…,10}, and A_(i−1) is a subset of A_(i) for each i between 2 and n, inclusive. For example, {}, {5,7}, {2,5,7}, {2,5,7}, {2,5,6,7,9} is one such sequence, with n=5.What is the remainder when K is divided by 10?

Original LaTeX notation

Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$.What is the remainder when $K$ is divided by $10$?

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_24
task_id: 8512ebae-eaa5-500a-9acf-4c1477055457
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2023~5fAMC~5f12A~5fProblems~2fProblem~5f24
task_revision_id: 2
config / split: default / train

Official source

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