Question:
5C1 + 5C2 + 5C3 + 5C4 +5C5 is equal to
(a) 30
(b) 31
(c) 32
(d) 33
Solution:
(b) 31
${ }^{5} C_{1}+{ }^{5} C_{2}+{ }^{5} C_{3}+{ }^{5} C_{4}+{ }^{5} C_{5}$
= ${ }^{5} C_{1}+{ }^{5} C_{2}+{ }^{5} C_{2}+{ }^{5} C_{1}+{ }^{5} C_{5}$
$=2 \times{ }^{5} C_{1}+2 \times{ }^{5} C_{2}+{ }^{5} C_{5}$
$=2 \times 5+2 \times \frac{5 !}{2 ! 3 !}+1$
$=10+20+1$
$=31$
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