Combinations

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Question

In a bag there are 5 red, 3 white and 4 black balls. Four balls are drawn from the bag. The number of ways in which at most 3 reel balls are selected is

Moderate

Solution

At most 3 red balls can be drawn in the following
ways
$\begin{array}{ccc} Number of red balls & Other colour balls & Number of ways \\ 0 & 4 & ^{7} C_{4} \\ 1 & 3 & ^{5} C_{1} \times^{7} C_{3} \\ 2 & 2 & ^{5} C_{2} \times^{7} C_{2} \\ 3 & 1 & ^{5} C_{3} \times^{7} C_{1} \end{array}$
Hence, required number of ways
$\begin{array}{l} =^{7} C_{4} +^{5} C_{1} \times^{7} C_{3} +^{5} C_{2} \times^{7} C_{2} +^{5} C_{3} \times^{7} C_{1} \\ = 35 + 175 + 210 + 70 = 490 \end{array}$

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Combinations

Remember concepts with our Masterclasses.

In a bag there are 5 red, 3 white and 4 black balls. Four balls are drawn from the bag. The number of ways in which at most 3 reel balls are selected is

At most 3 red balls can be drawn in the followingways Number of red balls Other colour balls Number of ways 04 7C413 5C1×7C322 5C2×7C231 5C3×7C1Hence, required number of ways=7C4+5C1×7C3+5C2×7C2+5C3×7C1=35+175+210+70=490

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