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  •  When , how many integral solutions exist for <img alt="\ma

 When \mathrm{a=2, b=3, c=4}, how many integral solutions exist for \mathrm{a+b+c+d=30} where \mathrm{a \geq 2, b \geq 3, a \geq 4} and \mathrm{d \geq 0}?

Option: 1

2024


Option: 2

1024


Option: 3

 264


Option: 4

428


Answers (1)

best_answer

\mathrm{a+b+c+d=30}\text{-Equation(1)}
\mathrm{a \geq 2, b \geq 3, c \geq 4}, and \mathrm{d \geq 0}

Which can also be written as \mathrm{a-2 \geq 0, b-3 \geq 0, c-4 \geq 0}, and \mathrm{d \geq 0}

\mathrm{x_{1}=a-2, x_{2}=b-3, x_{3}=c-4}
\mathrm{a=x_{1}+2, b=x_{2}+3, c=x_{3}+4 }
\mathrm{x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0, d \geq 0 }

Using it in equation (1)
\mathrm{x_{1}+x_{2}+x_{3}+d=21 }

Then the number of solutions for this will be \mathrm{={ }^{21+4-1} C_{4-1} }
                                                                      \mathrm{={ }^{24} C_{3} }
                                                                      \mathrm{=2024 }

Option (a) is correct.

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