equation involving factorials

(1) If $\left(x,y,z∈N+\left\{0\right\}\right)$ . Then the number of solutions of the equation $x!+y!=z!$ is?

(2) If $\left(x,y,z,t∈N+\left\{0\right\}\right)$ . Then the number of solutions of the equation $x!+y!+z!=t!$ is?

(3) If $\left(x,y,z,t,u∈N+\left\{0\right\}\right)$ . Then the number of solutions of the equation $x!+y!+z!=t!+u!$ is?

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

For number 1, the answer is 4. For (x, y, z): (0, 0, 2), (0, 1, 2), (1, 0, 2), and (1, 1, 2). Trivially.

For number 2, the answer is 2. For (x, y, z, t): (2, 2, 2, 3). Trivially.

For number 3,

John Ashley Capellan - 7 years, 6 months ago

1) assume x!>y! using this we get z! < 2(x!). but z is greater than x and we can prove for the sequence of factorials that : z! > 2(x!) if z>x>1 because even (n+1)! > 2(n!) for n>1. rest should be simple.

hemang sarkar - 6 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$