Combinatorics Exercises and German Grammar Notes

Combinatorics and Arithmetic Problems

• Exercise 8 (Zad 8):

  • Sub-task a: Calculation of total outcomes $N$ for two independent choices of $5$ options each:     $N = 5 \times 5 = 25$

  • Sub-task b: Calculation of total outcomes $N$ without repetition across two stages of $5$ and $4$ options:     $N = 5 \times 4 = 20$

• Exercise 9 (Zad 9):

  • Sub-task a: Product calculation:     $4 \times 3^3 = 128$

  • Supporting power calculations recorded in notes:     $4 \times 4 = 16$     $4 \times 4 \times 4 = 64$

  • Sub-task b: Multi-factor product evaluation:     $4 \times 3 \times 3 = 36$

• Exercise 10 (Zad 10):

  • Sub-task a: Product of equal factors:     $7 \times 7 = 49$     $N = 49$

  • Sub-task b: Sequential selection without replacement:     $7 \times 6 = 42$     $N = 42$

• Exercise 11 (Zad 11):

  • Sub-task a: Two-factor multiplication:     $12 \times 3 = 36$

  • Sub-task b: Three-factor permutation product:     $12 \times 3 \times 4 = 144$

  • Sub-task c: Four-factor permutation product:     $12 \times 3 \times 4 \times 5 = 720$

• Exercise 12 (Zad 12):

  • Sub-task a: Primary multiplication and adjustment step:     $N = 9 \times 2 = 18$     $18 - 1 = 17$

  • Sub-task b: Worked variants and strike-through revisions:

  • Primary formulation:       $N = 9 \times 9 \times 1 = 81$

  • Alternative products evaluated:       $9 \times 9 \times 2 = 162$       $8 \times 9 \times 2 = 144$       $9 \times 10 \times 2 = 180$       $9 \times 10 \times 1 = 90$

• Exercise 13 (Zad 13):

  • Sub-task a: Two-stage arrangement with $26$ possibilities each:     $N = 26 \times 26 = 676$

  • Sub-task b: Product of $10$ and $25$:     $N = 10 \times 25 = 250$

  • Sub-task c: Three-stage arrangements without repetition:     $N = 26 \times 25 \times 24 = 15600$

  • Sub-task d: Sequential product of descending integers:     $10 \times 9 \times 8 = 720$

  • Sub-task e: Selection combining $26$ letters and $10$ digits:     $N = 26 \times 10 = 260$

  • Sub-task f: Three-factor combination:     $26 \times 25 \times 10 = 6760$

• Exercise 14 (Zad 14):

  • Expression combining two pair products:     $12 \times 11 + 10 \times 9$

  • Step-by-step resolution:     $12 \times 11 = 132$     $10 \times 9 = 90$     $132 + 90 = 222$

• Exercise 15 (Zad 15):

  • Expression for sum of squared products:     $88 \times 88 + 27 \times 27$

  • Adjusted product sum:     $88 \times 87 + 27 \times 26$

• Marginal Scratchpad and Long Multiplication Work:

  • Left-column scratch calculations:     $81 \times 2 = 162$     $72 \times 2 = 144$

  • Right-margin vertical arithmetic steps:

  • Alphabet product step: $26 \times 26 = 676$

  • Partial product: $26 \times 25 = 650$

  • Extended product: $650 \times 24 = 15600$

  • Multiplication expansion of $88 \times 88$:       $88 \times 88 = 704 + 7040 = 7744$

  • Multiplication expansion of $27 \times 27$:       $27 \times 27 = 189 + 540 = 729$

  • Addition of squared terms:       $7744 + 729 = 8473$

  • Multi-digit vertical addition terms recorded in margin:       $1696196$       $15488$       $54208$       Final accumulated column sum: $5645376$

German Language and Grammar Notes

• Greeting and Inquiry Expressions:

  • German phrase structure: "Wie geht es …" accompanied by personal pronouns in the Dative case: (dir, ihm, ihr, ihnen).

  • Polish translation: "Co słychać u?" (Asking how someone is doing or what is happening with them).

• Dative Personal Pronoun Substitutions (Article to Pronoun Mapping):

  • Masculine/Neuter definite article dem maps to the masculine third-person singular Dative pronoun:     dem -> ihm

  • Feminine definite article der maps to the feminine third-person singular Dative pronoun:     der -> ihr

  • Plural definite article den (marked as L.mn for liczba mnoga / plural) maps to the third-person plural Dative pronoun:     den (L.mn) -> ihnen

• Inquiring About Health and Physical Ailments:

  • German question format: "Wem tut was weh?"

  • Polish contextual translation: "Komu co dolega" (Who is suffering from what / where does it hurt).

  • Model sentence structure for answering:     "Ihm tut das Bein weh" (His leg hurts / Pain is felt in his leg).

• Lesson Topic and Causal Clause Sentence Structure:

  • Lesson marker: "Stunde" (Lesson/Class unit).

  • Main lesson topic: "thema: Warum hast du Schnupfen?" (Topic: Why do you have a cold / runny nose?).

  • Causal clause construction using the subordinating conjunction "weil" (because):

  • Complete practice sentence: "Ich nehme Schmerztabletten, weil ich Kopfschmerzen habe." (I take pain tablets because I have a headache.)

  • Sentence stem prompt for further practice: "…, weil ich nicht" (…, because I do not)