True or False:
` "^(n)P_r` = `"^(n-1)P_(r-1)`, where r and n are natural numbers and n > 1

Prove that : ^nP_r= "^(n-1)P_r+r ^(n-1)P_(r-1) , for all natural numbers n and r for which the symbols are defined.

True or False: int x^ndx=(x^(n+1))/(n+1)+C , where n is any integer.

Prove that : ^nP_r= n"^(n-1)P_(r-1) , for all natural numbers n and r for which the symbols are defined.

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. If n is odd , the value of sum_(r-1)^(2) a_(2r -1) is

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. The value of sum_(r=0)^(n-1) a_(r) is

True or False : If A is any square matrix, then det (A^n) = (detA)^n where n is any natural number.

If X = {8^(n) -7n-1|n in N}Y = {49 (n-1)|n in N} where N is set of natural numbers , then :

If ""^(2n+1)P_(n-1): ""^(2n-1)P_n =3:5 then the value of n equals :

Verify that ""^nC_r= frac (n)(r) "^(n-1)C_(r-1) and hence prove that "^nC_r= (n!)/(r!(n-r)!) .