A bag contains 7 roses of which 5 red, one white and one yellow. Three roses are drawn at random without replacement. The probability that three roses have different colors’ is kl,(k,l)=1, then l−k=____

A bag contains 7 roses of which 5 red, one white and one yellow. Three roses are drawn at random without replacement. The probability that three roses have different colors’ is $\frac{k}{l}, (k, l) = 1, then l - k =____$

Solution:

All red roses are one unit

Total roses 7

Selecting 3 in $^{7} C_{3}$ ways

Selecting 3 different roses $^{3} C_{3} \times^{5} C_{1}$

Required probability $= \frac{^{3} C_{3} \times^{5} C_{1}}{^{7} C_{3}}$

$= \frac{1 \times 5}{\frac{7 \times 6 \times 5}{3 \times 2}}$

$= \frac{5}{7 \times 5}$

$\frac{k}{l} = \frac{1}{7}$

$l - k = 7 - 1 = 6$

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