In a shop there are five types of ice-creams available. A child buys six ice-creams.
Statement $-1 :$ The number of different ways the child can buy the six ice-creams is $^{10}C_5.$
Statement $-2 :$ The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $6 \,A's$ and $4 \,B's$ in a row.
Statement$-1$ is true, Statement$-2$ is false
Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is a correct explanation for Statement $-1$
Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is not acorrect explanation for Statement $-1$
Statement$-1$ is false, Statement$-2$ is true
The number of ways, in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together, is
In how many ways can $6$ persons be selected from $4$ officers and $8$ constables, if at least one officer is to be included
The number of words not starting and ending with vowels formed, using all the letters of the word $'UNIVERSITY'$ such that all vowels are in alphabetical order, is
In how many ways $5$ speakers $S_1,S_2,S_3,S_4$ and $S_5$ can give speeches one after the other if $S_3$ wants to speak after $S_1$ & $S_2$
If $^n{C_r} = {\,^n}{C_{r - 1}}$ and $^n{P_r}{ = ^n}{P_{r + 1}}$, then the value of $n$ is