Good day everyone.
I'm doing a combination problem and its solution keeps eluding me.
Problem:
Of a hand of 13 in a deck of 52 how many combinations are there of none of the cards being greater than 10, assuming that ace is greater than 10.
Attempted solution:
There are sixteen cards greater than 10: ace, jack, queen and king (all houses). So of thirteen cards in a house four are too great.
\begin{equation} \left( \begin{array}{c} 13 \\ 4 \end{array} \right) \end{equation}
Possible combinations of a hand consisting of only higher than 10 cards is:
\begin{equation} \left( \begin{array}{c} 16 \\ 13 \end{array} \right) \end{equation}
Multiply them together and subtract that from
\begin{equation} \left( \begin{array}{c} 52 \\ 13 \end{array} \right) \end{equation}
to get the wrong answer. There is something missing or wrong or both and I would appreciate help.
Thanks for your time.