Question

A child has a set of 26 alphabetized blocks and stacks 10 of these blocks in the shape of a pyramid.
The number of possible 10-block pyramids (as shown) that can be formed from the entire set of 26 blocks in an english alphabet is

• A. ${}^{26}{P}_4\cdot {}^{22}{P}_3\cdot {}^{19}{P}_2\cdot {}^{17}{P}_1$
• B. ${}^{26}{P}_4+{}^{22}{P}_3+{}^{19}{P}_2+{}^{17}{P}_1$
• C. ${}^{26}{C}_4\cdot {}^{22}{C}_3\cdot {}^{19}{C}_2\cdot {}^{17}{C}_1$
• D. ${}^{26}{C}_4+{}^{22}{C}_3+{}^{19}{C}_2+{}^{17}C$

Answer 1

Answer: (A)

We can arrange the first four blocks in ${}^{26}{P}_4$ ways, then the next three blocks in ${}^{22}{P}_3$ ways since only 22 blocks remain. Then we can arrange the next two in ${}^{19}{P}_2$ ways and the top block can be picked in ${}^{17}{P}_1$ ways. Since we are forming a single group (a pyramid) we multiply all the cases together.