Tamil Nadu Board of Secondary EducationHSC Commerce Class 11

A committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways this can be done when (i) atleast two ladies are included. (ii) atmost two ladies are included. - Business Mathematics and Statistics


A committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways this can be done when

  1. atleast two ladies are included.
  2. atmost two ladies are included.


(i) A committee of 5 is to be formed.

Possibilities Ladies (4) Gents (6) Combinations
1 2 3 `4"C"_2 xx 6"C"_3 = (4xx3)/(2xx1) xx (6xx5xx4)/(3xx2xx1)` = 120
  1 4 `4"C"_1 xx 6"C"_4 = 4 xx 6"C"_2 =
4 xx (6xx5)/(2xx1) = 60`
  0 5 4C0 × 6C5 = 1 × 6C1 = 1 × 6 = 6
      Total number of ways = 186

(ii) Almost two ladies are included means maximum of two ladies are included.

Possibilities Ladies (4) Gents (6) Combinations
1 2 3 `4"C"_2 xx 6"C"_3 = (4xx3)/(2xx1) xx (6xx5xx4)/(3xx2xx1)` = 120
2 3 2 `4"C"_3 xx 6"C"_2 = (4xx3xx2)/(3xx2xx1) xx (6xx5)/(2xx1)
 = 60`
3 4 1 4C4 × 6C1 = 1 × 6 = 6
      Total number of ways = 186
Concept: Combinations
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Chapter 2: Algebra - Exercise 2.4 [Page 38]


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