leehuan's last minute SOS thread (1 Viewer)

leehuan · Oct 20, 2015

Really just going to be perms and combs whatever I put here, anyway

Five letters are chosen from the letters of MOBILITY. These five letters are then placed alongside each other to form a five letter arrangement. How many arrangements are possible?

InteGrand · Oct 20, 2015

leehuan said:
Really just going to be perms and combs whatever I put here, anyway

Five letters are chosen from the letters of MOBILITY. These five letters are then placed alongside each other to form a five letter arrangement. How many arrangements are possible?

$In the word MOBILITY, the letter I appears twice, and the other letters (M,O,B,L,T,Y) each appear exactly once. We have two cases.$

$\underline{Case 1: 2 I's are chosen}.$

$In this case, we can choose where to put the I's first, in $\binom{5}{2}$ ways. We now have to choose and order three more letters from (M,O,B,L,T,Y), which can be done in $ ^6P _3$. Hence the no. of words in Case 1 is $\binom{5}{2}\times {^6}P _3$.$

$\underline{Case 2: No more than 1 I is chosen}.$

$This case is equivalent to forming 5 letter words from $\{M,O,B,I,L,T,Y\}$. This can be done in ${^7}P_5$ ways.$

$Hence the answer is $\binom{5}{2}\times {^6}P _3 + {^7}P_5$.$

leehuan · Oct 20, 2015

Hang on, having a mind blank. Why is it 5C2 for the arrangements of the two I's in Case 1 again?

InteGrand · Oct 20, 2015

leehuan said:
Hang on, having a mind blank. Why is it 5C2 for the arrangements of the two I's in Case 1 again?

The words consist of five letters: _ _ _ _ _.

Since in Case 1, we are having 2 I's, and I's are identical, we just choose the positions for them to go, which is done in ⁵C₂ ways (from the 5 spaces, choose 2 for the I's to occupy).

leehuan · Oct 20, 2015

InteGrand said:
The words consist of five letters: _ _ _ _ _.

Since in Case 1, we are having 2 I's, and I's are identical, we just choose the positions for them to go, which is done in ⁵C₂ ways (from the 5 spaces, choose 2 for the I's to occupy).

Oh of course. That was nice and easy enough; must've forgotten something. Thanks.

calamebe · Oct 20, 2015

leehuan said:
Hang on, having a mind blank. Why is it 5C2 for the arrangements of the two I's in Case 1 again?

We have 5 spots to put them, and we need to put 2 I's in the places, and the order does not matter as they are the same letter.

Nevermind beat me to it.

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