A box contains 15 cards numbered 1 to 15. Five cards are drawn at random one after the other with replacement. The numbers obtained from the cards drawn are noted. The probability that

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the smallest number is 9 is $\frac{16}{1215}$

the largest number is at most 7 is ${(\frac{7}{15})}^{5}$

the largest number is 7 is $\frac{7^{5} - 6^{5}}{15^{3}}$

the smallest number is at least 9 is ${(\frac{7}{15})}^{5}$

answer is A, B, C.

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Detailed Solution

a) Every number should be less than or equal to 7 and so ${(\frac{7}{15})}^{5}$
b) Every number should be less than or equal to 7 but all the numbers should not be less than 7 so that we get at least one 7 hence ${(\frac{7}{15})}^{5} - {(\frac{6}{15})}^{5}$ similar method for c & d