Find how many ways $10$ books can be arranged in a row on a shelf if:

the two oldest books must be in the middle two positions.

Find how many ways $10$ books can be arranged in a row on a shelf if:

the three newest books must not be separated.

Five cows and one set of twin calves can be housed separately in a row of seven stalls in ${}^{7}P_7=5040$ ways. Find in how many of these arrangements:

the two calves are not in adjacent stalls

Five cows and one set of twin calves can be housed separately in a row of seven stalls in ${}^{7}P_7=5040$ ways. Find in how many of these arrangements:

the two calves and their mother, who is one of the $5$ cows, are in adjacent stalls

Five cows and one set of twin calves can be housed separately in a row of seven stalls in ${}^{7}P_7=5040$ ways. Find in how many of these arrangements:

each calf is in a stall adjacent to its mother.

Find how many of the six-digit numbers that can be made from $1,2,2,3,3$ and $3:$

begin with a $2$

Find how many of the six-digit numbers that can be made from $1,2,2,3,3$ and $3:$

are not divisible by $2.$

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