A box contains 5 different red and 6 different white balls. In how many ways can 6 balls be selected so that there are at least

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A box contains 5 different red and 6 different white balls. In how many ways can 6 balls be selected so that there are at least two balls of each colour.

(a) 452

(b) 524

(c) 425

(d) 254

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(c) 425

6 balls consisting of at least two balls of each colour from 5 red and 6 white balls can be made in the following ways:

(a) Selecting 2 red balls out of 5 red balls and 4 white balls out of 6, i.e.,

Number of ways = 5C2 x 6C4$\frac{5\times4}{2}\times\frac{6\times5}{2}$

= 10 × 15 = 150

(b) Selecting 3 red balls out of 5 red balls and 3 white balls out of 6, i.e.,

Number of ways =  5C3 x 6C3$\frac{5\times4}{2}\times\frac{6\times5\times4}{3\times2}$

= 10 × 20 = 200

(c) Selecting 4 red balls out of 5 red balls and 2 white balls out of 6, i.e.,

Number of ways =  5C3 x 6C2$5\times\frac{6\times5}{2}$ = 5 × 15 = 75

∴ Total number of ways of selecting at least two balls of each colour = 150 + 200 + 75 = 425.