Find the number of all three elements subsets of the set`{a_1, a_2, a_3, ........... a_n}` which contain `a_3dot`

The number of all three element subsets of the set {a_(1),a_(2),a_(3)......a_(n)} which contain a_(3) , is

Find the number of ways of arranging 15 students A_1,A_2,........A_15 in a row such that (i) A_2 , must be seated after A_1 and A_2 , must come after A_2 (ii) neither A_2 nor A_3 seated brfore A_1

The number of all subsets of a set containing 2n+1 elements which contains more than n elements is

Let a_1=0 and a_1,a_2,a_3 …. , a_n be real numbers such that |a_i|=|a_(i-1) + 1| for all I then the A.M. Of the number a_1,a_2 ,a_3 …., a_n has the value A where : (a) A lt -1/2 (b) A lt -1 (c) A ge -1/2 (d) A=-2

A sequence of number is represented as a_1, a_2, a_3,….a_n . Each number in the sequence (except the first and the last) is the mean of the first two adjacent numbers in the sequece. If a_(1) = 1 and a_5 = 3 , what is hte value of a_(3) ?

If a_1,a_2,a_3,.......a_n , are 'n', distinct odd natural numbers, not divisible by any prime number greater than 5, then 1/a_1+1/a_2+1/a_3+......+1/a_n is less than (a) 15/8 (b) 17/8 (c)19/8 (d) 21/8

Statement 1: if A={2,4,6,},B{1,5,3}w h e r eAa n dB are the events of numbers occurring on a dice, then P(A)+P(B)=1. Statement 2: A_1, A_2, A_3,...... , A_n are all mutually exclusive events, then P(A_1)+P(A_2)+.......+P(A_n)=1.

If a_1, a_2, a_3 ......a_n (n>= 2) are real and (n-1) a_1^2 -2na_2 < 0 then prove that at least two roots of the equation x^n+a_1 x^(n-1) +a_2 x^(n-2) +......+a_n = 0 are imaginary.

.Let a_1, a_2,............ be positive real numbers in geometric progression. For each n, let A_n G_n, H_n , be respectively the arithmetic mean, geometric mean & harmonic mean of a_1,a_2..........a_n . Find an expression ,for the geometric mean of G_1,G_2,........G_n in terms of A_1, A_2,........ ,A_n, H_1, H_2,........,H_n .

rithmetic mean a, geometric mean G and Harmonic mean H to n positive numbers a_1,a_2,a_3,…..,a_n are given by A=(a_1+a_2+……………+a_n)/n, G=(a_1 a_2 a_n)^(1/2) and G= n/(1/H_1+1/H_2+………+1/H_n) There is a relation in A, G and H given by AgeGgeH equality holds if and only if a_1=a_2=..............=a_n On the basis of above information answer the following question If a,b,c are positive numers such that a+b+c=0 then greatest value a^3b^2c^5 is (A) 3^3.2^2.5^5 (B) 3^10 (C) 2^10 (D) 5^10