Question

(x+y)^15 бином! решите плз

Answer 1

Х+^15у. .............. ...

Answer 2

To expand the expression (x + y)^15 using the binomial theorem, we can use the formula:

(x + y)^n = Σ (n choose k) * x^(n-k) * y^k

where n is the power (in this case, 15), k is the index of the term (ranging from 0 to n), and (n choose k) is the binomial coefficient, which is calculated as:

(n choose k) = n! / (k! * (n - k)!)

Let's proceed to expand (x + y)^15 step by step:

1. Calculate the binomial coefficients for n = 15 and all possible k values (from 0 to 15):

(15 choose 0) = 15! / (0! * (15 - 0)!) = 1 (15 choose 1) = 15! / (1! * (15 - 1)!) = 15 (15 choose 2) = 15! / (2! * (15 - 2)!) = 105 (15 choose 3) = 15! / (3! * (15 - 3)!) = 455 (15 choose 4) = 15! / (4! * (15 - 4)!) = 1365 (15 choose 5) = 15! / (5! * (15 - 5)!) = 3003 (15 choose 6) = 15! / (6! * (15 - 6)!) = 5005 (15 choose 7) = 15! / (7! * (15 - 7)!) = 6435 (15 choose 8) = 15! / (8! * (15 - 8)!) = 6435 (15 choose 9) = 15! / (9! * (15 - 9)!) = 5005 (15 choose 10) = 15! / (10! * (15 - 10)!) = 3003 (15 choose 11) = 15! / (11! * (15 - 11)!) = 1365 (15 choose 12) = 15! / (12! * (15 - 12)!) = 455 (15 choose 13) = 15! / (13! * (15 - 13)!) = 105 (15 choose 14) = 15! / (14! * (15 - 14)!) = 15 (15 choose 15) = 15! / (15! * (15 - 15)!) = 1

2. Now, we can write out the expanded form of (x + y)^15:

(x + y)^15 = 1x^15y^0 + 15x^14y^1 + 105x^13y^2 + 455x^12y^3 + 1365x^11y^4 + 3003x^10y^5 + 5005x^9y^6 + 6435x^8y^7 + 6435x^7y^8 + 5005x^6y^9 + 3003x^5y^10 + 1365x^4y^11 + 455x^3y^12 + 105x^2y^13 + 15x^1y^14 + 1x^0y^15

Simplifying the coefficients:

(x + y)^15 = x^15 + 15x^14y + 105x^13y^2 + 455x^12y^3 + 1365x^11y^4 + 3003x^10y^5 + 5005x^9y^6 + 6435x^8y^7 + 6435x^7y^8 + 5005x^6y^9 + 3003x^5y^10 + 1365x^4y^11 + 455x^3y^12 + 105x^2y^13 + 15x^1y^14 + y^15

So, the expanded form of (x + y)^15 using the binomial theorem is:

x^15 + 15x^14y + 105x^13y^2 + 455x^12y^3 + 1365x^11y^4 + 3003x^10y^5 + 5005x^9y^6 + 6435x^8y^7 + 6435x^7y^8 + 5005x^6y^9 + 3003x^5y^10 + 1365x^4y^11 + 455x^3y^12 + 105x^2y^13 + 15x^1y^14 + y^15