ExERCISE A committee of four people is to be formed from a set of six superheroe

Question

ExERCISE A committee of four people is to be formed from a set of six superheroes, seven supervillains, and four ordinary citizens. How many ways are there to form the committee if (a) Captain Marvel (a superhero), must be on the committee. (b) At least two ordinary citizens must be on the committee. (c) At most one supervillain can be on the committee (d) At least one superhero and one supervillain must be on the committee, but Mr. Fantastic (a superhero) and Dr. Doom (a supervillain) cannot be on the committee together. or Clark Wayne (also ordinary citizens) are on it. Hint: note that these are all four of the ordinary citizens. the Joker (a supervillain). Nonetheless, at least one of these (identical) supervillains must be on the (e) Neither Bruce Allen nor Peter Kent (ordinary citizens) can be on the committee if either Barry Parker (f) The supervillains reveal that they are shapeshifters, and all transform into identical-looking copies of committee.

Explanation / Answer

Solution :

Given tha a committe should be formed from

6 superheroes

7 supervillians

4 ordinary citien

( a )

the total number of committees that include captain marvel = 16c3 = 560

( b )

the total number of committees that include atleast two citizen =  4c2 * 13c2 + 4c3 * 13c1 + 4c4

= 6 * 78 + 4 * 13 + 1

= 468 + 52 + 1 = 521

( c )

required that atmost one super villian can be on committe ., that is we have to add the number of committees with one super villian and committees with no super villian

the number of committees with one super villian = 7 * 10c3 = 840

the number of committees eith no super villian = 10c4 = 210

the number of committees with atmost one super villian = 1050

( d )

Here, Mr .fantastic and Dr Doom both should not be in a commettee together

the number of ways that contain Mr.Fantastic(Dr. Doom is excluded ) =  7c1 * 15c2 = 105 * 7 = 735

the number of ways that containDr. Doom ( Mr.Fantastic is excluded ) =  7c1 * 15c2 = 105 * 7 = 735

the number of committees containing aleast one superhero and one supervillian,

in which Mr.fentastic and Dr.Doom are not included = 735 + 735 = 1470

( e )

given that neither Bruce Allen nor Peter Kent can be on the commettee if Barry Parker or Clark Wayne

here let us consider these two pair of ordinary citizen as two groups where the members of one group cannot be accompanied by the members of other team members while forming a committee

the total number of committees formed by given condition =  15c4 + 15c4

= 2730

( f )

Given the super villians are shape shifters that means now these are all alike

=> the ways of sececting any number of super villians from the supervillians is always one

So, no let us calculate the number of ways of selecting a committee consisting atleast one super villian , which are alike , by adding the number of ways of selecting committee consisting 1,2,3,4 respectively

the number of ways of selecting a committee consisting one super villian = 1 * 10c3 = 120

the number of ways of selecting a committee consisting two super villian = 1 * 10c2 = 45

the number of ways of selecting a committee consisting three super villian = 1 * 10c1 = 10

the number of ways of selecting a committee consisting four super villian = 1

the total number of ways of selecting a committee consisting atleast one super villian = 120 + 45 + 10 + 1 = 176

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