A set of 12 cards contains 7 red and 5 black

Question

As the question follows,

In how many ways if we do not want $4$ cards of the same colour?

My solution so far

Total number of options of selecting 4 any cards: $495$

The number of options of $4$ of the same colour is: $175$

Therefore that means $495 - 175 = 320$ options that dont include $4$ cards in the same colour

My workings to get the answer above are:

$\dbinom{12}{4,8}$ Any selection of cards

$\dbinom74 \dbinom{5}{4}$ This is when there is either $4$ of red or $4$ of black.

Question

Would the answer $320$ be the correct answer?

Answer 1

There are only two colors, so you may first list all different size pairs first and then assign color for each of two stacks.
Consider stacks of size $3$ and $1$:
$(3, 1)$: ${7\choose 3}\cdot{5\choose 1} + {5\choose 3}\cdot{7\choose 1} = 35\cdot5+70=245.$
For this case: (3 red 1 black) + (3 black 1 red)

$(2, 2)$: ${7\choose 2}\cdot{5\choose 2} = 21\cdot 10 = 210$
So the total is $245+210=455.$

Answer 2

In how many ways can $4$ cards be selected from $7$ red and $5$ black cards if we do not want $4$ cards of the same colour?

Let's correct your solution.

There are $$\binom{12}{4}$$ ways to select four of the twelve cards.

From these, we must subtract those selections in which all of the selected cards are the same colour.

There are $$\binom{7}{4}$$ ways to select four of the seven red cards.

There are $$\binom{5}{4}$$ ways to select four of the five black cards.

Thus, the number of selections of four cards in which not all of the cards are of the same colour is $$\binom{12}{4} - \binom{7}{4} - \binom{5}{4} = 495 - 35 - 5 = 455$$

Your error: The number of ways of selecting four red or four black cards is found by adding the number of ways of selecting four black cards to the number of ways of selecting four red cards. $$\binom{7}{4} + \binom{5}{4}$$ Your calculation $$\binom{7}{4}\binom{5}{4}$$ counts the number of ways of selecting four red and four black cards, which is found by multiplying the number of ways of selecting four black cards by the number of ways of selecting four red cards.

When you see the word or, you should add. When you see the word and, you should multiply.