please explain because i am in learning stage .
A-1260 B-360 C-900 D-420
A-1260 B-360 C-900 D-420
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There are two vowels: O and A
There are five consonants: two Cs, two Ts, and one N
We're going to use all the letters in each arrangement, so each arrangement must be a 7-letter string.
Because the vowels must be separated, we should start by assigning their positions. There are
C(7,2) = 7 * 6 / 2 = 21 different sets of two positions we can use for the vowels. But 6 of these (any consecutive pair, from the first two to the last two) would put the vowels adjacent to each other, and they must be separate. So there are
21 - 6 = 15 pairs of positions for the vowels, O and A.
For each of these 15 pairs of positions, the vowels can be placed in 2 different orders. (The A can go in the first position and the O in the second one, or the O first and the A second.) So there are actually
15 * 2 = 30 ways to position the vowels.
That leaves the remaining 5 positions for the consonants. The N can go in any of those, so there are 5 possible positions for it.
Of the 4 positions not yet used, the Cs can go in any pair of them, and that will leave the other two for the Ts. So there are
C(4,2) = 4*3/2 = 6 ways to position the Cs and Ts, once the O, A, and N are placed.
Multiplying all these cases, we get a total of
30 * 5 * 6 = 900 possible arrangement (answer C).
I've described this in terms of assigning positions in the string to particular letters, taken in an order that I selected. But the way these calculations work, the result comes out the same if we take the letters in a different order. Choosing which letters to consider first is a matter of convenience; we can take whichever ones make the calculation easiest.
There are five consonants: two Cs, two Ts, and one N
We're going to use all the letters in each arrangement, so each arrangement must be a 7-letter string.
Because the vowels must be separated, we should start by assigning their positions. There are
C(7,2) = 7 * 6 / 2 = 21 different sets of two positions we can use for the vowels. But 6 of these (any consecutive pair, from the first two to the last two) would put the vowels adjacent to each other, and they must be separate. So there are
21 - 6 = 15 pairs of positions for the vowels, O and A.
For each of these 15 pairs of positions, the vowels can be placed in 2 different orders. (The A can go in the first position and the O in the second one, or the O first and the A second.) So there are actually
15 * 2 = 30 ways to position the vowels.
That leaves the remaining 5 positions for the consonants. The N can go in any of those, so there are 5 possible positions for it.
Of the 4 positions not yet used, the Cs can go in any pair of them, and that will leave the other two for the Ts. So there are
C(4,2) = 4*3/2 = 6 ways to position the Cs and Ts, once the O, A, and N are placed.
Multiplying all these cases, we get a total of
30 * 5 * 6 = 900 possible arrangement (answer C).
I've described this in terms of assigning positions in the string to particular letters, taken in an order that I selected. But the way these calculations work, the result comes out the same if we take the letters in a different order. Choosing which letters to consider first is a matter of convenience; we can take whichever ones make the calculation easiest.