Takrorlanmaydigan gruppalashlar. Chеkli to`plamlarning to`plam ostilari soni. Reja

Takrorlanmaydigan gruppalashlar. Chеkli to`plamlarning to`plam ostilari soni.
Reja:

1. 
   Paskal uchburchagi va N’yuton binomi
   
2. 
   Chekli to’plam qism to’plamlari soni.
   

1. Paskal uchburchagi va N’yuton binomi. 3°-xossaga ko’ra . Bundan 2° ga ko’ra ko’rinishdagi sonlarni Paskal uchburchagi ko’rinishida joylashtirish mumkin. Har bir son o’zining tepasidagi ikkita son yig’indisidan iborat.
.


Har bir qatordagi sonlar (a + b)^m ko’phadning yoyilmasidagi binomial koeffitsiyentlarga teng. Ularning yig’indisi m elementli X to’plamning barcha qism to’plamlari sonini beradi.
2.Chekli to’plam qism to’plamlari soni. 2 elеmеntli to‘plamning hammasi bo‘lib nechta qism to‘plami bоr degan savolga javob beraylik. Ular 1 ta bo‘sh, 2 ta 1 elеmеntli va 1 ta 2 elеmеntli, ya’ni to‘plamning o‘zidan ibоrat bo‘lgan qism to‘plamlardir. Jami: 1+2+1=4. Dеmak, 2 elеmеntli to‘plamning hammasi bo‘lib 4 ta qism to‘plami bоr ekan
Quvvati n ga teng bo’lgan A to’plamning to’plam ostilari soni 0 elementli, 1 elementli, 2 elementli, 3 elementli, …, n elementli toplam ostilari sonining yig’indisidan iborat bo’ladi.
Now consider the finite set S = {1, 2, 3, . . . , 8} (and so |S| = 8) and ask how many subsets (including S and the empty set ∅) are contained in S. As you might remember, there are 2⁸ such subsets, and this can be shown in at least two ways. The most direct way of seeing this is to form subsets of S
by the following process:

1	2	3	4	5	6	7	8
yes or no	yes or no	yes or no	yes or no	yes or no	yes or no	yes or no	yes or no

where in the above table, a subset if formed by a sequence of yes’s or
no’s according as to whether or not the corresponding element is in the subset. Therefore, the subset {3, 6, 7, 8} would correspond to the
sequence (no, no, yes, no, no, yes, yes, yes).
This makes it already clear that since for each element there are two choices (“yes” or “no”), then there must be
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁸possibilities in all.
Masalan А={1,2,3,4,5,6,7,8} to’plam quvvati |A|=8. To’plam ostilari soni 0 elementli, 1 elementli, 2 elementli, 3 elementli, 4 elementli, 5 elementli, 6 elementli, 7 elementli, 8 elementli toplam ostilari sonining yig’indisidan iborat
A to’plamning barcha qism to’plamlarini 0 va 1 lardan iborat ketma-ketlik bilan ifodalash mumkin. Agar element qism to’plamga tegishli bo’lsa, 1 bilan, tegishli bo’lmasa, 0 bilan almashtiramiz. Masalan {3,6,7,8} qism to’plamini (0,0,1,0,0,1,1,1) kabi shifrlash mumkin. Shunday kortejlar soni 2·2·2·2·2·2·2·2=2⁸ga teng.