A customer ordered fifteen Zingers. Zingers are placed in packages of four, three, or one. In how many different ways can this order be filled?

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A customer ordered fifteen Zingers. Zingers are placed in packages of four, three, or one. In how many different ways can this order be filled?

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Answer 1 · 2017-02-12T11:29:20

$15$ $15$

Explanation:

There are at most $3$ $3$ packages of four Zingers. Treating each possibility $3, 2, 1, 0$ $3, 2, 1, 0$ as a separate case, there are then subcases according to the number of packages of three Zingers.

It's probably best to just systematically enumerate them as follows:

$o o o o o o o o o o o o o o o o o o$ $o o o ocolor(white)(o)o o o ocolor(white)(o)o o o ocolor(white)(o)o o o$
$o o o o o o o o o o o o o o o o o o o o$ $o o o ocolor(white)(o)o o o ocolor(white)(o)o o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$

$o o o o o o o o o o o o o o o o o o o$ $o o o ocolor(white)(o)o o o ocolor(white)(o)o o ocolor(white)(o)o o ocolor(white)(o)o$
$o o o o o o o o o o o o o o o o o o o o o$ $o o o ocolor(white)(o)o o o ocolor(white)(o)o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$
$o o o o o o o o o o o o o o o o o o o o o o o$ $o o o ocolor(white)(o)o o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$

$o o o o o o o o o o o o o o o o o o o o$ $o o o ocolor(white)(o)o o ocolor(white)(o)o o ocolor(white)(o)o o ocolor(white)(o)ocolor(white)(o)o$
$o o o o o o o o o o o o o o o o o o o o o o$ $o o o ocolor(white)(o)o o ocolor(white)(o)o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$
$o o o o o o o o o o o o o o o o o o o o o o o o$ $o o o ocolor(white)(o)o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$
$o o o o o o o o o o o o o o o o o o o o o o o o o o$ $o o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$

$o o o o o o o o o o o o o o o o o o o$ $o o ocolor(white)(o)o o ocolor(white)(o)o o ocolor(white)(o)o o ocolor(white)(o)o o o$
$o o o o o o o o o o o o o o o o o o o o o$ $o o ocolor(white)(o)o o ocolor(white)(o)o o ocolor(white)(o)o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$
$o o o o o o o o o o o o o o o o o o o o o o o$ $o o ocolor(white)(o)o o ocolor(white)(o)o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$
$o o o o o o o o o o o o o o o o o o o o o o o o o$ $o o ocolor(white)(o)o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$
$o o o o o o o o o o o o o o o o o o o o o o o o o o o$ $o o ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$
$o o o o o o o o o o o o o o o o o o o o o o o o o o o o o$ $ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)ocolor(white)(o)o$

I count $15$ $15$ arrangements.

Answer 2 · 2017-02-12T14:18:28

See below.

Explanation:

Calling ${alpha,beta,gamma} in ZZ ge 0$ we have

$3alpha+2beta+gamma=15$

The different solutions to this equation, calling diophantine equation after Diophantus of Alexandria, equals the number of different arrangements.

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A customer ordered fifteen Zingers. Zingers are placed in packages of four, three, or one. In how many different ways can this order be filled?

2 Answers

Explanation:

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