Permutations & Combinations
Integer solutions with lower bound constraints
nta_pyq_2023_jan
Grade 11
Question:
Number of integral solutions to the equation x + y + z = 21, where x $\geq$ 1, y $\geq$ 3, z $\geq$ 4, is equal to _____.
Step-by-Step Solution
Key Concept: Substitute x'=xā1, y'=yā3, z'=zā4 to reduce to x'+y'+z'=13 with x',y',z'$\geq$0. Use stars and bars.
After substitution: x'+y'+z'=13, non-negative integers. Number of solutions = \(\binom{13+2}{2} = \binom{15}{2} = \frac{15 \times 14}{2} = 105\).
Correct Answer: 105