Using combination

Chapter 16 Class 11 Probability
Concept wise

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Misc 2 4 cards are drawn from a well-shuffled deck of 52 cards. What is the probability of obtaining 3 diamonds and one spade? Total 4 cards are to be selected out of 52 cards Hence Total ways to select 4 cards = 52C4 We need to select 3 diamond card & 1 spade card Total number of diamond card = 13 Number of diamond card to be selected = 3 Total ways to select 3 diamond card = 13C3 Also, Total number of spade card = 13 Number of spade card to be selected = 1 Total ways to select 1 spade card = 13C1 Hence, Probability of obtaining 3 diamonds and one spade = 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔 𝒕𝒐 𝒔𝒆𝒍𝒆𝒄𝒕 𝟑 𝒅𝒊𝒂𝒎𝒐𝒏𝒅 × 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔 𝒕𝒐 𝒔𝒆𝒍𝒆𝒄𝒕 𝟏 𝒔𝒑𝒂𝒅𝒆﷮𝑻𝒐𝒕𝒂𝒍 𝒘𝒂𝒚𝒔 𝒕𝒐 𝒔𝒆𝒍𝒆𝒄𝒕 𝟒 𝒄𝒂𝒓𝒅𝒔﷯ = 𝟏𝟑𝑪𝟑 × 𝟏𝟑𝑪𝟏﷮𝟓𝟐𝑪𝟒﷯ = 13!﷮3!(13 − 3)﷯ × 13!﷮1! 13 −1﷯!﷯﷮ 52!﷮4! 52 − 4﷯!﷯﷯ = 13!﷮3!10!﷯ × 13!﷮1!12!﷯﷮ 52!﷮4!48!﷯﷯ = 13!﷮3!10!﷯ × 13!﷮12!﷯ × 4!48!﷮52!﷯ = 286﷮20825﷯