Question

3. Асия собрала цветы: 9 ромашек и 12 незабудок. Сколькими способами можно составить букет из 8 Цветов, если: а) в букете 6 ромашки и 4 незабудки; b) в букете как минимум должны быть 3 незабудки​

Answer 1

Ответ:

а) 41 580

б) 197 505

Объяснение:

р - ромашки, н - незабудки

а) нам надо 6р из 9 и 4н из 12

1) 6р из 9 это $C^{6}_{9} =\frac{9!}{6!(9-6)!} =\frac{6!*7*8*9}{6!*3!} =\frac{7*8*9}{1*2*3} =84$

2) 4н из 12 это (аналогично) = $C^{4}_{12} = \frac{12!}{4!8!} = 495$

3) в букете должны быть И ромашки, И незабудки, поэтому умножаем

84*495=41 580 способов

б) аналогично а)

в букете могут быть либо 3н и 5р либо 4н и 4р либо 5н и 3р либо 6н и 2р либо 7н и 1р либо 8н и 0р

$C^{3}_{12} *C^{5}_{9} +C^{4}_{12} *C^{4}_{9} +C^{5}_{12} *C^{3}_{9} +C^{6}_{12} *C^{2}_{9} +C^{7}_{12} *C^{1}_{9} +C^{8}_{12} *C^{0}_{9} =$ (также через факториалы) = 220*126 + 495*126 + 792*84 + 924*36 + 792*9 + 495*1 = 197 505 способов

Answer 2

Problem Analysis

To solve this problem, we need to find the number of ways to arrange the flowers in a bouquet given certain conditions. Let's analyze each condition separately:

a) In the bouquet, there should be 6 daisies and 4 forget-me-nots. b) In the bouquet, there should be at least 3 forget-me-nots.

Solution

a) To find the number of ways to arrange the flowers in the bouquet with 6 daisies and 4 forget-me-nots, we can use the concept of combinations. The formula for combinations is given by:

C(n, r) = n! / (r! * (n-r)!)

where n is the total number of flowers and r is the number of daisies.

In this case, n = 10 (6 daisies + 4 forget-me-nots) and r = 6 (number of daisies). Plugging these values into the formula, we get:

C(10, 6) = 10! / (6! * (10-6)!) = 210

Therefore, there are 210 ways to arrange the flowers in the bouquet with 6 daisies and 4 forget-me-nots.

b) To find the number of ways to arrange the flowers in the bouquet with at least 3 forget-me-nots, we can use the concept of combinations again. We need to consider two cases:

Case 1: Exactly 3 forget-me-nots In this case, we have 3 forget-me-nots and 5 remaining flowers (6 daisies - 3 forget-me-nots). Using the formula for combinations, we get:

C(8, 5) = 8! / (5! * (8-5)!) = 56

Case 2: More than 3 forget-me-nots In this case, we have 4 forget-me-nots and 4 remaining flowers (6 daisies - 4 forget-me-nots). Using the formula for combinations, we get:

C(8, 4) = 8! / (4! * (8-4)!) = 70

To find the total number of ways, we add the number of ways from each case:

56 + 70 = 126

Therefore, there are 126 ways to arrange the flowers in the bouquet with at least 3 forget-me-nots.

Answer

To summarize: a) There are 210 ways to arrange the flowers in the bouquet with 6 daisies and 4 forget-me-nots. b) There are 126 ways to arrange the flowers in the bouquet with at least 3 forget-me-nots.

Please let me know if you need any further clarification.