Question

The number of non-negative integral solutions to the system of equations $x+y+z+u+t=20$ and $x+y+z=5$

• A. $336$
• B. $346$
• C. $246$
• D. None of these

Answer 1

The correct option is A $336$

Given, $x+y+z+u+t=20$...........(1)

and $x+y+z=5$................(2)

The given system of equations can be written as

$u+t=15$........(3)

$x+y+z=5$.............(4)

No. of non-negative integral solutions of (4) are ${{}^{n+r-1}c}_r={{}^{3+5-1}c}_5={{}^{7}c}_5$

No. of non-negative integral solutions of (3) are ${{}^{n+r-1}c}_r={{}^{2+15-1}c}_{15}={{}^{16}c}_{15}$

So, required numbers $={{}^{16}c}_{15}\times {{}^{7}c}_5=336$