For Auyona

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Problem 01 · Combinatorics

The Memory Wall

Yash is building a memory wall for Auyona to celebrate their favorite moments together. He has collected 7 photographs, each representing a different special memory they share — their first date, a favorite trip, a meaningful conversation.

However, the wall only has space for 4 photographs. Since Yash wants the display to read like a timeline from left to right, the order of the photographs matters.

In how many distinct ways can Yash choose and arrange 4 of the 7 memories on the wall?

7P4  =  7! / (7 − 4)!  =  ?
That's our P — for Permutations.

Problem 02 · Statistics

A Quiet Regression

Yash, ever the data scientist, has been quietly recording two things over the past five months: x = months since their first date, and y = the number of flowers he has brought Auyona that month.

x12345
y23578

Compute Pearson's correlation coefficient r, rounded to two decimal places.

Nearly perfect — that's our R, for Pearson's r. The trend, it turns out, is well-modeled by a line.

Problem 03 · Hypothesis Testing

The Playlist's Promise

Yash is making a playlist for Auyona and tells her that he designed it to match her favorite music balance: 25% rap, 20% house, 30% indie, and 25% pop. To check whether the playlist actually follows that distribution, Auyona randomly samples 80 songs from the playlist and records each song's genre. Her results:

GenreRapHouseIndiePop
Observed24122816

At the α = 0.05 significance level, is there convincing evidence that the genre distribution of Yash's playlist is different from the distribution he claimed?

Under H0, expected counts are: Rap 20, House 16, Indie 24, Pop 20.
χ² = (24−20)²/20 + (12−16)²/16 + (28−24)²/24 + (16−20)²/20 ≈ 3.27.

With df = 3, the critical value at α = 0.05 is 7.815. Since 3.27 < 7.815, we fail to reject H0. Yash's claim holds — there isn't convincing evidence the playlist differs from what he promised.

And there is your O — the one nested inside H0.

Problem 04 · Linear Algebra

The Hidden Note

Yash creates a digital scrapbook for Auyona filled with photos from their favorite dates together. On the final page, he hides a love note behind a password. To unlock it, Auyona must find the determinant of the matrix below, where each row represents one of their special memories:

M  =  213 102 411

What is the determinant of matrix M, and what number should Auyona enter to reveal Yash's message?

That's your M, the matrix itself. The proof is complete.