Probability of Receiving 2-4-6-7-Queen-King Sequence from 40 Cards

[tomwilliam]

Homework Statement

I'm helping my son with past papers before an exam - I know the answer to this but don't know why.

Relevant Equations

combinatorics equations

A quick translation of the question: We have a deck of 40 cards, containing four suits (hearts, diamonds, spades, clubs), in which each suit has an ace, a queen, a jack, a king and six number cards (2 to 7). From the deck, six cards are distributed randomly and successively to a player who picks them up in the order he receives them. What is the probability of obtaining the sequence 2 - 4 - 6 - 7 - queen - king in the cards he receives.

My thinking is that the P = number of favourable outcomes / total universe of possible outcomes.

So looking first at the favourable outcomes: the top of that fraction should be a sequence of any of the four 2s, then any of the four 4s, etc. until we reach six cards. There are four suits, so that should be 4 x 4 x 4 x 4 x 4 x 4 = 4^6.

The bottom half should be simply any 6 cards taken from a set of 40, so that would be ^40 A_6 (where I think the term A (Arrangement) might be P = Permutation in English).

So my answer would be (A). It turns out the answer is (C)... so where did I go wrong?

Thanks in advance!

 

[PeroK]

The answer C assumes that the cards are all the same pre-specified suit. For example, the probability of being dealt the 2-4-6-7-D-R of clubs (in that order) equals $(\frac 1 {40})(\frac 1 {39}) \dots (\frac 1 {35})$, which is answer C.

For the question as you have interpreted it, the answer is A.

 

[tomwilliam]

The answer C assumes that the cards are all the same pre-specified suit. For example, the probability of being dealt the 2-4-6-7-D-R of clubs (in that order) equals $(\frac 1 {40})(\frac 1 {39}) \dots (\frac 1 {35})$, which is answer C.

For the question as you have interpreted it, the answer is A.

Thank you! I considered this possibility, but discarded it on the following logic:

If the question means that the sequence has to be all of the same suit, it should still be a factor of 4 in answer (C), as there are four possible suits for it to work with.

If the question wants a specific suit, it doesn't appear to say that anywhere, which I think is pretty misleading.

Thanks for your help!