Simple. (Problem $9$ )

Algebra Level pending

$x^{(10)} =$ ?

Give your answer as $a + b + c + d + e + f + g + h + i + j$ where $a, b, c, d, e, f, g, h, i, j$ is the coefficients of the equation.

Note: $x^{(n)}$ is the rising factorial.

Bonus: In the discussion, see if you can simplify it to the form $y! \pm a$ or $y!$ where $a, y$ are positive integers.

The answer is 3628800.

1 solution

Yajat Shamji
Jul 12, 2020

$x^{(n)} = x(x + 1)(x + 2)(...)(x + n - 1)$

Since $n = 10$ :

$x^{(10)} = x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9)$

$x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9) = x^{10} + 45x^9 + 870x^8 + 9450x^7 + 63273x^6 + 269325x^5 + 723680x^4 + 1172700x^3 + 1026576x^2 + 362880x$

Since the answer is supposed to be given in the form of $a + b + c + d + e + f + g + h + i + j$ :

$1 + 45 + 870 + 9450 + 63273 + 269325 + 723680 + 1172700 + 1026576 + 362880 = \fbox{3628800}$

Bonus: $3628800 = \fbox{10!}$

Simple. (Problem 9 9 9 )

The answer is 3628800.

1 solution

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Simple. (Problem $9$ )