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

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

x + y + z = 5                                        (2)

Given, system of equations is equivalent to

x + y + z = 5                                       (3)

and, u + t = 15                                    (4)

Number of non-negative integral solutions of equation (3)

${=}^{\mathrm{n}+\mathrm{r}-1}{\mathrm{C}}_{\mathrm{r}}{=}^{3+5-1}{\mathrm{C}}_5{=}^7{\mathrm{C}}_5$,

Number of non-negative integral solutions of equation (4)

${=}^{\mathrm{n}+\mathrm{r}-1}{\mathrm{C}}_{\mathrm{r}}{=}^{2+15-1}{\mathrm{C}}_{15}{=}^{16}{\mathrm{C}}_{15}$

∴ Required number ${=}^7{\mathrm{C}}_5{\times }^{16}{\mathrm{C}}_{15}=336$.