Sinfda 30ta o’quvchi bor. Ularning 9 tasining uchtadan do’sti bor, 11tasining – to’rttadan,
10tasining – beshtadan do’sti bor. Bunday bo’lishi mumkinmi?

Mazkur masalani yechish uchun, biz grafiklar nazariyasidan foydalanamiz. Bu masalada o'quvchilarni grafning uchlari (cho'qqilari) sifatida, do'stlikni esa grafning qirralari (chetlari) sifatida tasavvur qilish mumkin.

1. **Birinchi qadam:** O’quvchilar soni \( n = 30 \).
2. **Ikkinchi qadam:** Har bir o'quvchining do'stlari sonini yig'indisini topamiz:
   - 9 ta o'quvchi uchtadan do'sti bor: \( 9 \times 3 = 27 \)
   - 11 ta o'quvchi to’rttadan do'sti bor: \( 11 \times 4 = 44 \)
   - 10 ta o'quvchi beshtadan do'sti bor: \( 10 \times 5 = 50 \)
   - Umumiy yig'indi: \( 27 + 44 + 50 = 121 \)

3. **Uchinchi qadam:** Grafikda qirralar sonini hisoblash. Har bir qirra ikki uchta mavjud (ya'ni, do'stlik ikki tomonlama):
   - Do'stliklar yig'indisi \( 121 \) bo'lsa, qirralar soni \( \frac{121}{2} \) bo'lishi kerak.
   - Biroq 121 soni 2 ga bo’linmaydi, shuning uchun grafik mavjud bo’lishi mumkin emas.

**Xulosa:** Bunday holatning grafikda bo'lishi mumkin emas. Bu shuni anglatadiki, berilgan sharoitda o'quvchilar orasidagi do'stlik aloqalari mavjud bo'lishi mumkin emas.

Do you want details or have any questions? 

Here are some related questions:
1. How can graph theory be used to solve other problems like this?
2. What is the significance of degree sum in graph theory?
3. Can you give examples of real-life applications of this type of graph theory problem?
4. How would the scenario change if there were an even total number of friendships?
5. What if there were an additional condition, like some students not having any friends?

**Tip:** Graph theory is powerful in solving problems where relationships or connections between objects are involved, making it useful in social networks, computer networks, and more.

Explore a mathematical problem using graph theory to analyze friendships among students. Understand how to apply graph theory concepts to determine possible relationships among students based on given conditions.

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