Permutations and Combinations Test - 4

The number of arrangements of n different things taken r at a time which include a particular thing is r ^n-1P_r-1 = rP(n-1,r-1)

The number of arrangements of n different things taken r at a time which exclude a particular thing is ^n-1P_r  = P(n-1,r)

Since they are boarding from ground floor and we are considering the number of ways they leave the lift ,we can consider there are 7 floor sas we exclude the ground floor)

As each of the 5 persons can leave the lift in 7 ways, required number of ways= 7⁵ 

We have  nC_{r - 1}  + ⁿC_r = ⁿ⁺¹C_r

Hence ¹²C₄ + ¹²C₅ = ¹³C₅ ................(i)

But given ¹²C₄ + ¹²C₅ =ⁿC₅ ................(ii)

Comparing (i) and (ii) we get  n = 13

each time there are 6 possibilities, therefore for n times there are 6ⁿ  possibilities.

there are 4 boys and 4 girls and the row can start either with a boy or girl, therefore the number of ways are 4! x 4! x 2

there are four consonants and 3 vowels, therefore 3! * 4!

you can either choose true or false, therefore for 10 questions you will have 2¹⁰ possibilities.