Volleynomial

The required polynomial is $P(x) \;= \; x + \tfrac{7}{15}(x-3)(x-4)(x-5)(x-6)(x-7) \; = \; x + 56\binom{x-3}{5}$ making the answer $123 + 56\binom{120}{5} = \boxed{10672369467}$ .

As $P(n)$ is a 5th degree polynomial, $P(n)-n$ is also a 5th degree polynomial.

$P(n)-n$ has roots $n=3,4,5,6,7$ . So it can be written as $P(n)-n=k(n-3)(n-4)(n-5)(n-6)(n-7)$

To find value of $k$ , We have been given $P(8)=64$

$64-8=k(5)(4)(3)(2)(1)$

$k=\frac{7}{15}$

Now

$P(123)-123=(120)(119)(118)(117)(116)*\frac{7}{15}$

Thus we have $P(123)=123+22869362880*\frac{7}{15}=123+10672369344=10672369467$

Polynomial $p(n)$ is of the form:

$\begin{aligned} P(n) & = {\color{#3D99F6}A}(n-3)(n-4)(n-5)(n-6)(n-7) + n & \small \color{#3D99F6} \text{where }A \text{ is a constant.} \\ P(8) & = A(5)(4)(3)(2)(1) + 8 = 64 & \small \color{#3D99F6} \implies A = \frac 7{15} \\ \implies P(n) & = \frac 35 (n-3)(n-4)(n-5)(n-6)(n-7) - n \\ P(123) & = \frac 7{15} \times 120 \times 119 \times 118 \times 117 \times 116 + 123 \\ & = \boxed{10672369467} \end{aligned}$