Lucky Number

Level 2

We call a number lucky if the sum of its digits are $7$ . Let $2014$ be the $N$ th lucky number .

Find the $\frac{N}{6}$ th lucky number .

Source: BdMO 2014

The answer is 124.

2 solutions

Jubayer Nirjhor
Feb 8, 2014

Balls in urns / Stars and bars yields a total of $\dfrac{8\cdot 9}{2}+ \dfrac{7\cdot 8}{2}=36+28=64$ non-negative integral solutions to the two equations below:

$a+b+c=7~~~~~~~~~~~~~~~~~1+a+b+c=7$

Hence there are $64$ lucky numbers in $[1,1999]$ . So $2014$ is the $66$ th lucky number. Again Stars and bars yields $8$ non-negative integral solutions to $a+b=7$ , so we need to find the $\dfrac{66}6 - 8=3$ rd lucky number greater than $99$ which is trivial.

The answer is $\fbox{124}$ .

Wanchun Shen
Feb 8, 2014

Of course it's not a good solution...

7 16 25 34 43 52 61 70 106 115

124 133 142 151 160 205 214 223 232 241

250 304 313 322 331 340 403 412 421 430

502 511 520 601 610 700 1006 1015 1024 1033

1042 1051 1060 1105 1114 1123 1132 1141 1150 1204

1213 1222 1231 1240 1303 1312 1321 1330 1402 1411

1420 1501 1510 1600 2005 2014

Hence, N=66, the 11th number is 124.

The number of lucky numbers from $1$ to $999$ is the number of integral solutions to the equation $x+y+z=7$ with $0\le x,y,z\le 9$ which is $36$ using stars and bars.The number of 4-digit lucky numbers with the first digit equal to $1$ is the number of non-negative solutions to the equation $x+y+z=6$ which is 28.Therefore,there are $36+28=64$ lucky numbers between $1$ and $1999$ . Therefore, $2005$ must be the $65$ th lucky number and $2014$ must be the 66th lucky number.Therefore, $N=66$ .The final answer is derived by listing out the first $11$ values which is an easy task.

Rahul Saha - 7 years, 4 months ago

Lucky Number

The answer is 124.

2 solutions

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