Question

Suppose a photographer is arranging 6 out of a group of 9 inhabitants of the Island of Knights & Knaves for a group photo.

Of these 9 inhabitants, 5 are knights, namely A, B, C, D, and E.
Of these 9 inhabitants, the other 4 are knaves, namely W, X, Y, and Z.

a. How many ways can the photographer choose 6 of these 9 inhabitants to be in the photo?

b. How many ways can the photographer choose 6 of these 9 inhabitants to be in the photo so that the number of knights in the photo is equal to the number of knaves?

c. How many ways can the photographer choose 6 of these 9 inhabitants to be in the photo so that the number of knights in the photo is greater than or equal to the number of knaves?

d. How many ways can the photographer choose 6 of these 9 inhabitants to be in the photo so that the number of knights in the photo is not greater than or equal to the number of knaves?

e. How many ways can the photographer choose 6 of these 9 inhabitants to be in the photo so that A is in the photo or all 4 knaves are in the photo? (this is an inclusive or)

f. How many different ways can the photographer arrange 6 of these 9 inhabitants in a line?

Answer 1

Hello

a) Number of ways photographer can arrange 6 out of 9 individuals is

b) Number of ways photographer can arrange 6 out of 9 individuals such that number of knights and knaves are equal=

c) Number of ways photographer can arrange 6 out of 9 individuals such that number of knights are greater than or equal to knaves are =

d) Number of ways photographer can arrange 6 out of 9 individuals such that number of knights are not greater than or equal to knaves are=

e) Number of ways photographer can arrange 6 out of 9 inhabitants such that A is in photo or all the knaves are in photo =

f) Number of ways of arranging 6 of 9 individuals in a line = 9*8*7*6*5*4 = 60480 ways

I hope this solves your doubt.

Do upvote if you find this helpful.