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  • There are _______ three-digit numbers with digit sums of 15.Option: 1 120Option: 2 <

There are _______ three-digit numbers with digit sums of 15.

Option: 1

120


Option: 2

119


Option: 3

118


Option: 4

 117


Answers (1)

best_answer

Assuming the 3-digit number as xyz.

Given that sum of the digits of the number is 15
Then \mathrm{x+y+z=15}
 

Two digits can be zero, but one of them has to be greater than 1 .
Assuming that \mathrm{x \geq 1, y \geq 0} and \mathrm{z \geq 0}

Again assuming that \mathrm{z \geq 0}

Again assuming that \mathrm{x-1=s}
\mathrm{x=1+s}

Then \mathrm{1+s+y+z=15}
\mathrm{s+y+z=14}

And
\mathrm{s \geq 0, y \geq 0} and \mathrm{z \geq 0}
Then the number of solutions will be
\mathrm{={ }^{14+3-1} C_{3-1}}
\mathrm{= { }^{16} C_{2}}
\mathrm{=120}

But \mathrm{x=15} is not possible for \mathrm{s=14}

The total solutions will be \mathrm{=120-1}
                                         \mathrm{=119}

The correct option is (b)

Posted by

shivangi.shekhar

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