Find the number of distinct arrangements that can be made from all the letters in the word THEATRE when the arrangement:

begins with two $Ts$ and ends with two $Es$

Find the number of distinct arrangements that can be made from all the letters in the word THEATRE when the arrangement:

has $H$ as its middle letter

Find the number of distinct arrangements that can be made from all the letters in the word THEATRE when the arrangement:

ends with the three vowels $E, A$ and $E.$

The following diagram shows a row of post boxes with the owners' names beneath. Five parcels, one for the owner of each box, have arrived at the post office. If one parcel is randomly placed in each box, find the number of ways in which: 

the five parcels can all be placed in the correct boxes

The following diagram shows a row of post boxes with the owners' names beneath. Five parcels, one for the owner of each box, have arrived at the post office. If one parcel is randomly placed in each box, find the number of ways in which: 

exactly one parcel can be placed in the wrong box

The following diagram shows a row of post boxes with the owners' names beneath. Five parcels, one for the owner of each box, have arrived at the post office. If one parcel is randomly placed in each box, find the number of ways in which: 

the correct parcels can be placed in $\mathrm{Mr}$ $A\text{'}s$ and one other person $\text{'}s$ box only

The following diagram shows a row of post boxes with the owners' names beneath. Five parcels, one for the owner of each box, have arrived at the post office. If one parcel is randomly placed in each box, find the number of ways in which: 

exactly two parcels can be placed in the correct boxes.

Find how many permutations there are of:

five from seven distinct objects