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Question

If 4- digits number greater that 5000 are randomly formed the digits 0,1,3,5 and 7, then what is the probability of forming a number divisible by 5, when

(i) the digits may be repeated?

(ii) the repetition of digits in not allowed?

Or

A card is drawn from a deck of 52 cards . Find the probability of getting

(i) an ace.

(ii) a king or a heart or a red card.

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Solution

(i) let 4-digits number be formed as

I II III IV

I first place can be filled in two ways. using digit 5 or 7 .[ number greater than 5000]

Each of the remaining three places can be filled in 5 ways.

Total numbers so formed =2×5×5×5=250

If the number is divisible by 5. then IV place can be filled in 2 ways using digit 0 or 5.

Number, so formed that are divisible by 5

2×5×5×2=100

Hence, required probability =Favourable casesTotal cases

=100250=25

(ii) Let 4 places in a 4-digits number be formed as I II III IV

I place can be filled in two ways, using digit 5 or 7.

Then places II , III,IV may be filled,

=4×3×2

Total number of are so formed =2×4×3×2

n(S)=48ways.

If the number is divisible by 5, o or 5 is placed at unit place, ie. at IV place.

IIIIIIIV50

IIIIIIIV50

IIIIIIIV75

In each case, places II and III can be filed in 3×2.i.e, 6 ways.

n(E)=6+6+6=18

Hence , required probability =n(E)n(S)=1848=38

Or

(i) There are four aces in a pack of 52 cards, out of which one ace card can be drawn in 4C1ways.

Favorable number of outcomes =4C1=4

So, required probability=452=113

(ii) There are 26 red card,out of which one red card can be drawn in 26C1=26

Favorable number of number of outcomes =26C1=26

So, required probability =2652=12

(iii) There are 26 red cards including 13 hearts plus 2 red king and there are 2 more kings,. Therefore there are 28 ways cards which are either red or king or heart , out of which one card can be drawn in 26C1=26 ways,

Favorable number of outcomes =28

Each of 52 cards, one card can be drawn in 52C1=52 ways.

Total number of outcomes =52C1=52

So, required probability =2852=713


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