A box contains 5 different red and 6 different white balls. In how man

1.	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
Answer» (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.

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

Answer»

(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

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.