Question

There are 26 letters in the English alphabet. Letters are picked one by one at random, so that each letter has the same chance of appearing as any other letter, regardless of which letters have appeared or not appeared.
i) Ten (10) letters are picked. Find the chance that the sequence that appears is RANDOMNESS, in that order.
ii) Find the chance that they can be arranged to form the word RANDOMNESS.
My solution approach is as follow:
P((A) that the sequence that appears RANDOMNESS in that order)=(10/26)*(.../...)*(1/17)=
(10/26)*(9/25)*(8/24)*(7/23)*(6/22)*(5/21)*(4/20)*(3/19)*(2/18)*(1/17)=
ii) P((A) can be arranged to form the word RAMDOMNESS)= 10C1/26C10 = 10/26C10=

Answer 1

the chance of appearing any letter is same regardless of which letters have appeared or have not. I think your solution is wrong^^

Answer 2

For the former, it's relatively simple:
Note that the probability of picking the correct letter, in order, is \(\frac{1}{26}\), also note that RANDOMNESS is 10 letters long, hence:
\[
\left(\frac{1}{26}\right)^{10}=P(A)
\]Where \(A\) is the event picking the exact letters, in order.
Interestingly enough, there's not much need for the latter problem. Note that the total possible number of rearrangements in RANDOMNESS is \(10!\), so, noting that we could get any of the one rearrangements from any of the \(26^{10}\) possible solutions, multiplying by the permutations gives us the probability of the favourable cases. Hence:
\[
\left(\frac{1}{26}\right)^{10}\cdot10!=P(B)
\]Where \(B\) is the probability of picking the exact letters of RANDOMNESS in any permutation.

Answer 3

What does this notation stand 10! means ....

Answer 4

can you quantify 10! as

Answer 5

Sorry, mate. We have \(10!=10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1\).

Answer 6

Thanks! ubderstood