How many ways can you arrange the letters in the word Toronto if you must begin with a T and end in an O?
Fill in the restrictions first: we have 2 Ts, and 3 Os, let's assume they are all
different, e.g. a red T and a blue T
TxxxxxO
2xxxxx3
2*5*4*3*2*1*3
but the Os are indistinguishable as are the Ts, so we divide by 3! and 2!
so 2*5*4*3*2*1*3/(3!2!)
= 60
630
To find the number of ways you can arrange the letters in the word "Toronto" while starting with a T and ending with an O, follow these steps:
Step 1: Determine the total number of letters in the word "Toronto". In this case, there are 7 letters.
Step 2: Since the first letter must be a T and the last letter must be an O, we can fix these two letters in their respective positions. This leaves us with 5 remaining letters to arrange.
Step 3: The remaining 5 letters can be arranged in $(5!)$ ways.
Step 4: Multiply the number of ways we can arrange the remaining 5 letters by the number of ways we can fix the T and O in their positions.
$(5!)\times2! = 5\times4\times3\times2\times1\times2\times1 = 240$
Therefore, there are 240 ways to arrange the letters in the word "Toronto" if you must begin with a T and end in an O.
To find the number of ways you can arrange the letters in the word "Toronto" if you must begin with a T and end with an O, we can follow these steps:
1. Identify the number of letters in the word "Toronto." In this case, we have 7 letters.
2. Since we must begin with a T and end with an O, we fix these two positions. So we are left with 5 remaining positions to arrange the remaining 5 letters.
3. Calculate the number of ways to arrange the remaining unoccupied positions using the remaining letters. There are 5 letters remaining (r, o, n, t, and o) to be arranged in the 5 remaining positions.
4. To calculate the number of arrangements, we can use the concept of permutations. The number of ways to arrange 5 items in 5 positions can be found using the formula for permutations: n! / (n - r)!, where n is the total number of items and r is the number of positions to be filled.
Applying this formula:
Number of arrangements = 5! / (5 - 5)!
= 5! / 0!
= 5! / 1
= 5!
Therefore, there are 5! (5 factorial) arrangements or 120 ways to arrange the letters in the word "Toronto" if you must begin with a T and end with an O.