What is the probability that the deal of a five-card

Explanation:
Method 1
If we start with a full deck there are 52 cards; 4 aces; 48 non-aces
Card 1 deal should not be an Ace (48 non aces; 52 cards)
P(Card 1 not Ace)${\displaystyle =\frac{48}{52}}$
Card 2 deal should not be an Ace (47 non aces; 51 cards)
P(Card 2 not Ace)${\displaystyle =\frac{47}{51}}$
Card 3 deal should not be an Ace (46 non aces; 50 cards)
P(Card 3 not Ace)${\displaystyle =\frac{46}{50}}$
Card 4 deal should not be an Ace (45 non aces; 49 cards)
P(Card 4 not Ace)${\displaystyle =\frac{45}{49}}$
Card 5 deal should not be an Ace (44 non aces; 48 cards)
P(Card 5 not Ace)${\displaystyle =\frac{44}{48}}$
So then:
P(all five cards non aces)${\displaystyle =\frac{48}{52}\cdot \frac{47}{51}\cdot \frac{46}{50}\cdot \frac{45}{49}\cdot \frac{44}{48}}$
${\displaystyle =\frac{35673}{54145}}$
=0.658841...
Method 2
Using the combination formula:
${\displaystyle {}_{\left\{n\right\}}{C}^r=(beg\in \left\{array\right\}\left\{c\right\}n\mathrm{\setminus }rend\left\{array\right\})=\frac{n!}{r!(n-r)!}}$
The number of combination of choosing five non-aces from 52 cards (48 of which are not aces) is given by:
n(all five cards non aces)${\displaystyle {=}_{48}{C}^5=(beg\in \left\{array\right\}\left\{c\right\}48\mathrm{\setminus }5end\left\{array\right\})}$
${\displaystyle =\frac{48!}{5!(48-5)!}}$
${\displaystyle =\frac{48!}{5!43!}}$
And the total number of all combinations of choosing any five cards
n(any five cards) ${\displaystyle {=}_{52}{C}^5=(beg\in \left\{array\right\}\left\{c\right\}52\mathrm{\setminus }5end\left\{array\right\})}$
${\displaystyle =\frac{52!}{5!(52-5)!}}$
${\displaystyle =\frac{52!}{5!47!}}$
P (all five cards non aces)${\displaystyle =\frac{n\left(allfivecardsnonaces\right)}{n\left(anyfivecards\right)}}$
${\displaystyle =\frac{\frac{48!}{5!43!}}{\frac{52!}{5!47!}}}$
${\displaystyle =\frac{48!}{5!43!}\cdot \frac{5!47!}{52!}}$
${\displaystyle =\frac{47!\cdot 48!}{43!\cdot 52!}}$
${\displaystyle =\frac{43!\cdot 44\cdot 45\cdot 46\cdot 47\cdot 48!}{43!\cdot 48!\cdot 49\cdot 50\cdot 51\cdot 52}}$

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