Question

How many integral solutions are there to x + y + z + = 29, when  1, y≥ 2, z ≥ 3 and t ≥ 0?

Solution

Correct option is

2600

We have,

             1, y ≥ 2, z ≥ 3 and t ≥ 0, where x, y, z, t are integers

Let      u = x – 1, v = y – 2, w = z – 3.

Then,  ≥ 1  ⇒  u 0  ;  y ≥ 2   ⇒   u ≥ 0  ;  z ≥ 3  ⇒  w ≥ 0

Thus, we have

           u + 1 + v + 2 + w + 3 + t = 29  ⇒  + v + w + t = 23                 [where   u  0 ; v ≥ 0 ; w  0]                                                       

The number of solution of above equation is equal to number of ways to divide 23 identical objects among 4 groups such that gets 0 or more.

⇒    The total number of solutions of this equation is

       

SIMILAR QUESTIONS

Q1

The sides AB, BC and CA of a triangle ABC have a, b and c interior points on them respectively, then find the number of triangles that can be constructed using these interior points as vertices.

Q2

Let is a set containing n elements. A subset P of set X is chosen at random. The set X is then reconstructed by replacing the elements of set Pand another set Q is chosen at random then find the number of ways to form sets such that 

Q3

Let is a set containing n elements. A subset P of set X is chosen at random. The set X is then reconstructed by replacing the elements of set Pand another set Q is chosen at random. Find  number of ways to  chosenand Q such that ∪ Q contains exactly r elements.

Q4

In how many ways can 12 books be equally distributed among 3 students?

Q5

10 different toys are to be distributed among 10 children. Total number of ways of distributing these toys so that exactly 2 children do not get any toy, is equal to:

Q6

There are 20 persons among whom are two brothers. Find the number of ways in which we can arrange them around a circle so that there is exactly one person between the two brothers.

Q7

In how many ways can a party of 4 men and 4 women be seated at a circular table so that no two women are adjacent?

Q8

There are 5 gentlemen and 4 ladies to dine at a round table. In how many ways can they seat themselves so that no two ladies are together?

Q9

Find the number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated.

Q10

How many integral solutions are there to the system of equations

x1 + x2 + x3 + x4 + x5 = 20 and x1 + x2 + x3 = 5 when xk ≥ 0?