How many integer solutions to ₁+₂+3+4 = 28 are there for which #₁ ≥ 6, ₂ ≥ 6, 13 ≥ 3 and 4 ≥ 1?

Answered: How many integer solutions to ₁ + x₂ +…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

How many integer solutions to ₁+₂+3+4 = 28 are there for which #₁ ≥ 6, ₂ ≥ 6, 13 ≥ 3 and 4 ≥ 1?

Expert Solution

Step 1: solving the problem

$g i v e n space e q u a t i o n space i s space x subscript 1 plus x subscript 2 plus x subscript 3 plus x subscript 4 equals 28........................ left parenthesis 1 right parenthesis space a n d space c o n d i t i o n s space a r e space g i v e n space x subscript 1 greater or equal than 6 comma x subscript 2 greater or equal than 6 comma x subscript 3 greater or equal than 3 comma x subscript 4 greater or equal than 1 W e space r e d u c e space t h e space p r o b l e m space t o space o n e space i n space t h e space n o n minus n e g a t i v e space i n t e g e r s. space S i n c e space x subscript 1 space space i s space a n space i n t e g e r comma S i m i l a r l y comma space sin c e space x subscript 2 space space a n d space x subscript 3 space comma x subscript 4 space a r e space i n t e g e r s comma$

$\begin{aligned} y_{1} & = x_{1} - 6 \\ y_{2} & = x_{2} - 6 \\ y_{3} & = x_{3} - 3 \\ y_{4} & = x_{4} - 1 \end{aligned}$

$\begin{aligned} y_{1} + 6 + y_{2} + 6 + y_{3} + 3 + y_{4} + 1 & = 28 \\ ⟹ y_{1} + y_{2} + y_{3} + y_{4} & = 28 - 16 \\ ⟹ y_{1} + y_{2} + y_{3} + y_{4} & = 12. . . . . . . . . . . . . . . . . . . . . . . (2) \end{aligned}$