DisMat Bonus

Show that any nonnegative integer $t$ , we have the identity

(0 8 t) - (2 8 t) + (4 8 t) - ... + (8 t 8 t) = 1 6^{t}

(1 + x)^{8 t} + (1 - x)^{8 t} = 2 k = 0 \sum n (k 8 t) x^{k}

1 + (2 8 t) + (8 8 t) + (16 8 t) + ... + (4 t 8 t) = 1 6^{t}