The total number of 9-digit numbers which have all different digits is
The number of words which can be formed out of the letters of the word 'ARTICLE', so that vowels occupy the even place is
Given 5 different green dyes, four different blue dyes and three different red dyes, the number of combinations of dyes which can be chosen taking atleast one green and one blue dye is
If ${ }^n P_r=840$ and ${ }^n C_r=35$, then $r$ is equal to ............ .
Given that, ${ }^n P_r=840$ and ${ }^n C_r=35$
$$\begin{array}{ll} \because\quad { }^n P_r={ }^n C_r \cdot r! \\ \Rightarrow \quad &840=35 \times r! \\ \Rightarrow & r!=\frac{840}{35}=24 \\ \Rightarrow & r!=4 \times 3 \times 2 \times 1 \\ \Rightarrow & r!=4! \\ \Rightarrow & r=4 \end{array}$$
${ }^{15} C_8+{ }^{15} C_9-{ }^{15} C_6-{ }^{15} C_7$ is equal to ............ .
$$\begin{aligned} { }^{15} C_8+{ }^{15} C_9-{ }^{15} C_6-{ }^{15} C_7 & ={ }^{15} C_{15-8}+{ }^{15} C_{15-9}-{ }^{15} C_6-{ }^{15} C_7 \quad\left[\because{ }^n C_r={ }^n C_{n-r}\right] \\ & ={ }^{15} C_7+{ }^{15} C_6-{ }^{15} C_6-{ }^{15} C_7 \\ & =0 \end{aligned} $$