Question

out of 7 gents and 4 ladies a committee of 5 is to be formed. the number of committees such that each committee includes at least one lady is ​

Answer 1

1 st ans is wrong right answer is 336

your questions says as at least 3 ladies it means minimum 3 ladies in committee

so we have 4 ladies so 3 or 4 ladies are selecting in committee is valid..

so we have become combination ...

⁸c4×⁴c3 or ⁸c3×⁴c4

(⁸c4×⁴c3) + (⁸c3×⁴c4)

solve it ..

280+56=336

100 percent sure ans...like it..

Answer 2

The number of committees such that each committee includes at least one lady is 441

Given :

Out of 7 gents and 4 ladies a committee of 5 is to be formed

To find :

The number of committees such that each committee includes at least one lady

Solution :

Step 1 of 2 :

Calculate total number of ways of selection

Here it is given that out of 7 gents and 4 ladies a committee of 5 is to be formed

Total number of members = 7 + 4 = 11

Now a committee of 5 is to be formed

∴ Total number of ways of selection

$\displaystyle \sf{ = {}^{11} C_5 }$

$\displaystyle \sf{ = \frac{11!}{5! \: (11 - 5)!} }$

$\displaystyle \sf{ = \frac{11!}{5! \times 6!} }$

$\displaystyle \sf{ = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6!}{(5 \times 4 \times 3 \times 2) \times 6!} }$

$\displaystyle \sf{ = \frac{11 \times 10 \times 9 \times 8 \times 7 }{(5 \times 4 \times 3 \times 2) } }$

$\displaystyle \sf{ = \frac{11 \times 9 \times 8 \times 7 }{( 4 \times 3 ) } }$

$\displaystyle \sf{ = 11 \times 3 \times 2 \times 7 }$

$\displaystyle \sf{ = 462}$

Step 2 of 2 :

Calculate the number of committees such that each committee includes at least one lady

The number of committees such that each committee includes no lady

$\displaystyle \sf{ = {}^{7}C_5 }$

$\displaystyle \sf{ = \frac{7!}{5! \: (7 - 5)!} }$

$\displaystyle \sf{ = \frac{7!}{5! \times 2!} }$

$\displaystyle \sf{ = \frac{7 \times 6 \times 5!}{5! \: \times 2} }$

$\displaystyle \sf{ = 21 }$

Hence the number of committees such that each committee includes at least one lady

= 462 - 21

= 441

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