The variables x1 x2 x3 and x4 are integers that must satisfy

The variables x1, x2, x3, and x4 are integers that must satisfy the given conditions. Write out the generating functions for the number of solutions of this equation. Write your answer as a quotient in which the denominator is a power of (1-x) and the numerator has at most 5 terms.

x1 + x2 + x3 +x4 = n

5 ? x1 ? 14

10 ? x2

0 ? x3 ? 9

5 ? x4

? = less than or equal t0

Solution

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The variables x1, x2, x3, and x4 are integers that must satisfy the given conditions. Write out the generating functions for the number of solutions of this equation. Write your answer as a quotient in which the denominator is a power of (1-x) and the numerator has at most 5 terms.

x1 + x2 + x3 +x4 = n

5 ? x1 ? 14................WHAT ARE ALL THESE QUESTION MARKS ? ..THE POST IS NOT CLEAR TO RESPOND PROPERLY ..

PLEASE COPY & PASTE OR WRITE IN WORDS THE PROBLEM TO GET PROPER RESPONSE ...

10 ? x2

0 ? x3 ? 9

5 ? x4

? = less than or equal t0

GIVING YOU THE METHODOLOGY , ASSUMING AS FOLLOWS

The variables x1, x2, x3, and x4 are integers that must satisfy the given conditions. Write out the generating functions for the number of solutions of this equation. Write your answer as a quotient in which the denominator is a power of (1-x) and the numerator has at most 5 terms.

x1 + x2 + x3 +x4 = n

5 < = x1 < = 14

10 < = x2

0 < = x3 < = 9

5 < = x4

? = less than or equal t0..........IGNORING THIS .............

LET US INTRODUCE SOME NEW VARIABLES AS FOLLOWS TO MAKE ALL CRITERIA AS ....Xi > = 0 ....

PUT ...

Y1 = X1- 5 ........Y1 > =0

Y2=X2-10....Y2>=0

......Y3=X3 > = 0

.Y4=X4-5..............Y4>=0

SO THE PROBLEM IS NOW REDUCED TO FINDING THE NUMBER OF SOLUTIONS FOR ...

Y1+5+Y2+10+Y3+Y4+5

Y1+Y2+Y3+Y4+Y5 > = N - 20

ASSUMING N IS GREATET THAN > =20 , WE CAN INTERPRET THE ABOVE AS ...

THE COEFFICIENT OF X^(N-20) IN THE PRODUCT ...

[Y1^0+Y1^1+Y1^2+...+Y1^19]*[Y2^0+Y2^1+Y2^2+....]*[Y3^0+Y3^1+....Y3^9]*[Y4^0+Y4^1+.....]*[Y5^0+Y5^1+...]........

THE ABOVE CAN BE WORKED OUT USING BINOMIAL EXPN . FORMULAS THEN ..

IF YOU POST THE PROBLEM PROPERLY WE CAN WORK OUT THE SOLUTION CORRECTLY