wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Let x+y+z+w=30; where x,y,z,wN. If m and n denote the number of solutions when no variable may exceed 10 and each variable is an odd number respectively, then m+n=

Open in App
Solution

When no variable may exceed 10
The number of solutions = coefficient of x30 in
(x+x2+x3+...+x10)4
= coeff. of x30 in [x(1x10)1x]4
= coeff. of x30 in x4(1x10)4(1x)4
= coeff. of x30 in x4(14x10+6x20...)(1x)4
= coeff. of x30 in (x44x14+6x24)(1x)4

Now, coeff. of x26 in (1x)4=29C3
coeff. of x16 in (1x)4=19C3
coeff. of x6 in (1x)4=9C3

The number of required solutions is,
m=29C3419C3+69C3=282

When each variable is an odd number
put x=2n+1,y=2p+1,z=2q+1,w=2r+1
x,y,z,w1 & x+y+z+w=30
(2n+1)+(2p+1)+(2q+1)+(2r+1)=30
n+p+q+r=13 & n,p,q,r0
The number of required solutions is,
= 13+41C41=16C3=560

m+n=842

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Multinomial Theorem
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon