we need to find the number of positive integer solutions for following equation
we first found number of integers then positive
for any doubt
please comment
1. Fix n and k. How many positive integer solutions are there to x1 + +...
For a given k, 1 ≤ k ≤ n, how many vectors (x1, x2, . . . , xk) are there for which each xi is a positive integer such that 1 ≤ x1 < x2 < · · · < xk ≤ n?
For a given k, 1 ≤ k ≤ n, how many vectors (x1, x2, . . . , xk) are there for which each xi is a positive integer such that 1 ≤ x1 < x2 < · · · < xk ≤ n?
Problem 3 Let n and k > l be positive integers. How many different integer solutions are there to x1 +...+ In = k, with all xi <l?
How many integer solutions are there for the inequality : x1 +
x2 + x3 + x4 ≤ 15
(a) if xi ≥ 0
(b) if 6 ≥ x1 ≥ 1, 6 ≥ x2 ≥ 1, x3 ≥ 0, x4 > 0
How many integer solutions are there for the inequality : x++ (a) if z 20
How many integer solutions are there for the inequality : x++ (a) if z 20
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
How many non-negative integer solutions are there to the following problem? x1 + x2 + x3 = 10 where x1 >= 2
Determine the number of integer solutions of x1 + x2 + x3 + x4-32, where a) xi 2 0, 1 3is4 b) x1, x2 2 2, x3, X4 2 1
discrete math, use the the formula on the paper
m) (C5) How many integer solutions are there to the equation x + y + z = 8 for which... (a) x,y,z are all positive? ht(k-1 (b) x,y,z are all nonnegative? (c) x, y, z are all greater than -3? 1 (balls + boxes-1 I boxes- I +(**) - (saldo
. The two questions that follow concern the following variant of the Bernoulli pro- cess: Fix k 2 1. At each (integer) time n 2 1 the process takes the value Xn, where Xn are i.i.d. random variables each with the uniform distribution on 12,,. (4) (a) What is the PMF for the random variable N defined as the smallest N 2 2 so that XN X1 (b) Is N a stopping time? (c) What is the probability that XN+1...
How to solve these problem, I need detailed answer process.
7. a) What is the multinomial coefficient of x1x2'- ...xx*, ii + /2ikn, in the expansion of (x1+ x2 + .. + xk)"? b) How many terms are there of the form the multinomial (xi + x2 xx)? i. i.i.x1'x212 ...xxk in the expansion of 1 L2.k
7. a) What is the multinomial coefficient of x1x2'- ...xx*, ii + /2ikn, in the expansion of (x1+ x2 + .. + xk)"?...