10-hand card is dealt from a well shuffled deck of 52 cards A 10-hand card is dealt from a well shuffled deck of 52 cards. What is the probability that the hand contains at least two cards from each of the four suits?

You either need the suits distributed $4222$ or $3322$.  The chance of $4222$ is $$\frac{{4 \choose 1} (\text{suit with four cards}) {13 \choose 4}{13 \choose 2}^3}{52 \choose 10}$$  The chance of $3322$ is $$\frac{{4 \choose 2} (\text{suits with three cards}) {13 \choose 3}^2{13 \choose 2}^2}{52 \choose 10}$$ for a total of $\displaystyle\frac {7592832}{27657385} \approx 0.2745$
