Question

The number of positive integral solutions to the
system of equations a1 + a2 + a3 + a4 + a5 = 47 and
a1+a3 = 37 is
(a) 2376
(b) 2246
(c) 2024
(d) 1296​

Answer 1

Given : a1 + a2 + a3 + a4 + a5 = 47 and  a1+a3 = 37 is

To Find :  number of positive integral solutions

(a) 2376

(b) 2246

(c) 2024

(d) 1296​

Solution:

a1 + a2 + a3 + a4 + a5 = 47  

a1+a3 = 37

=> a2 + a4 + a5 = 10

a2 =1     This mean  a4 + a5  = 9 can be in 8 ways

a2 =2     This mean  a4 + a5  = 8 can be in 7 ways

a2 =8    This mean a4 + a5 = 2 can be in 1 way

ways = 8 + 7 + .................+ 1

= 8 * 9/2

= 36 Ways

  a1+a3 = 37

a1 can be from 1 to 36    

=> 36 Ways

Total  ways

36 * 36  =  1296 ways  

The number of positive integral solutions to the  system of equations a1 + a2 + a3 + a4 + a5 = 47 and  a1+a3 = 37 is 1296

option D is correct

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