how to do this one?

let s be the set of integers x such that x=a^3+b^3+c^3-3abc,for some integers a,b,c.prove that if x,y are included in s,then xy is included in s

consider the 2 degree polynomial with integer coefficients P(x) = a+ bx + cx²

so x= P(1)P(w)P(w²)
and y = Q(1)Q(w)Q(w²)
so xy can be written as G(1)G(w)G(w²) where G(x) = P(x)Q(x)

degree of G(x) is atmost 4

we can write G(x) = (x³ - 1)A(x) + R(X) degree of R(x) is atmost 2

so G(x) - R(x) has roots as 1,w,w²

so xy can be written as R(1) R(w) R(w²)

so its included

1)

p/n

prob that he will die in a year is p

and prob that he will be first to die is 1/n

so prob that he will die ina year and will be first to die is p/n

an easy way to test whether this answer is correct is to see what happens if p>1/n.

The number of people who die in that year is not n, so the prob of A₁ being the first is not 1/n

reviving this!

@kunl ,,

Yaar statements bhi copy.!! :P (for 'reviving this')

Lagta hai main famous ho gaya hoo!! :P

suppose k people die , probability of this is ⁿC_kp^k(1-p)^n-k

you want A1 to be among them , probability of this is ^n-1C_k-1ⁿC_k

among that p that A1 dies first is 1/k

so ans = Σ ⁿC_kp^k(1-p)^n-kkn1k = 1/n ?