Constant and degrees of freedom of a two-sample t statistic

Let $X_1,X_2,X_3,Y_1,Y_2,Y_3,Y_4$ be i.i.d. $N(\mu,\sigma^2)$ random variables. Let $\overline X=\frac13\sum_{i=1}^3X_i$ and $\overline Y=\frac14\sum_{j=1}^4Y_j$. If $k\sqrt{\frac{15}7}\dfrac{(\overline X-\overline Y)}{\sqrt{\left\{\sum_{i=1}^3(X_i-\overline X)^2+\sum_{j=1}^4(Y_j-\overline Y)^2\right\}}}$ has $t_\nu$ distribution, then $(\nu-k)$ equals __________________